Critical Problem of Optimal Stabilization Without Control Constraints

Abstract

This paper investigates the linear–quadratic optimal stabilization problem in the so-called critical case, that is, the situation in which the spectrum of the system matrix contains purely imaginary eigenvalues or the standard positive-definiteness conditions on the weight matrices of the objective functional are violated. To address these challenges, new regularization methods for critical problems via perturbation of the system matrices and the functional are studied, and novel algorithms for decomposing multidimensional problems into a collection of one-dimensional canonical systems are developed. The main contribution of this work lies in providing a systematic framework for critical cases where standard methods fail. The results are of practical significance for the construction of optimal synthesis in various engineering and applied systems; in particular, they are applicable to the stabilization of unmanned aerial vehicles, robotic complexes, and intelligent power grids.

1. Introduction

The problem of finding optimal controllers and achieving optimal stabilization within the linear–quadratic regulator (LQR) framework is among the most fundamental and practically relevant problems in optimal control theory. At the theoretical level, this problem is reduced, via Bellman’s dynamic programming method, to the construction of a linear feedback controller, which in turn requires solving the algebraic matrix Riccati equation [1,2,3]. The practical solvability of this equation depends strongly on the spectral properties of the system matrix and on the dimension of the state space.

Optimal stabilization problems have been studied extensively for processes of a diverse nature, including systems with distributed parameters [4,5,6], systems subject to impulsive and stochastic disturbances [7,8,9], and others, with the linear–quadratic formulation serving as the theoretical foundation for these investigations. Moreover, LQR-based methods are widely employed in the mathematical modeling of complex processes in aerospace engineering, robotics, and energy systems, and constitute a standard approach to the stabilization of unmanned aerial vehicles (UAVs), including quadcopters [10,11,12,13,14]. Furthermore, sophisticated methods of mathematical modeling and stability analysis find crucial applications across various disciplines to address complex real-world dynamics, ranging from mechanical systems to recent stochastic socio-economic and epidemiological models [15,16].

The main obstacle to using LQR is the need for an accurate linear model of the object. Such models are obtained either by linearizing complex nonlinear systems, or using various integrations of LQR with artificial intelligence methods and control of complex network objects. In recent years, there has been a new wave of interest in linear–quadratic problems due to their combination with artificial intelligence methods. Current research focuses on “safe reinforcement learning” (Safe RL), where LQR acts as a stabilizing framework for neural network controllers. In addition, data-based control methods (model-free LQR) are actively developing, which allow the synthesis of optimal strategies without exact knowledge of the object matrices [11,17,18].

Despite significant advances, the analysis of current critical operating regimes remains a challenging mathematical problem. To address these challenges, this paper provides a systematic framework for linear–quadratic optimal stabilization in the critical case, specifically when the standard positive-definiteness conditions on the weight matrices are violated or the system matrix contains purely imaginary eigenvalues.

The main contribution and novelty of this work lie in the development of new regularization methods via precise perturbations of the system matrices and the objective functional, yielding solutions in extended function spaces where the classical formulation admits no admissible controls. Furthermore, we develop novel algorithms for decomposing multidimensional critical problems into a collection of independent one-dimensional canonical systems.

2. Problem Formulation

Following [1,2,3], the classical optimal stabilization problem is formulated as follows: find the optimal control in feedback (synthesis) form

$$u=u\left[x\right(t\left)\right],$$

(1)

where the function $x\left(t\right):R\to R^n$ describes the controlled process and satisfies the Cauchy problem

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left[x\left(t\right)\right],0=t_0<t\le \infty ,$$

(2)

$$x\left(0\right)=x^0,$$

(3)

and the objective functional

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(x^T\left(t\right)Qx\left(t\right)+u^T\left(t\right)Hu\left(t\right)\right)dt$$

(4)

attains its minimum value. Here the vector $x^0\in R^n$ is fixed but arbitrary.

In problem (1)–(4), $A,B,Q,H$ are matrices of dimensions $n\times n$, $n\times r$, $n\times n$, and $r\times r$, respectively; the matrices Q and H are symmetric and positive definite; the pair $(A,B)$ is completely controllable, i.e., the Kalman rank condition holds

$$rang\left(B,AB,A^{2}B,\dots ,A^{n-1}B\right)=n.$$

The solution of the optimal stabilization problem is sought in the following class of admissible pairs:

$$\mathrm{Y}(x,u)\equiv \left\{\left(x\left(t\right),u\left(t\right)\right):x\left(t\right)\in W_2^{1,n}(0,\infty ),u\left(t\right)\in L_2^r(0,\infty )\right\},$$

(5)

where

$$L_2^r(0,\infty )\equiv \left\{u\left(t\right):{\int }_0^{\infty }{\displaystyle \sum _{j=1}^r}{u}_j^2\left(t\right)dt<\infty \right\},$$

$$W_2^{1,n}(0,\infty )\equiv \left\{x\left(t\right):{\int }_0^{\infty }{\displaystyle \sum _{j=1}^n}{x}_j^2\left(t\right)dt<\infty ,{\int }_0^{\infty }{\displaystyle \sum _{j=1}^n}{\dot{x}}_j^2\left(t\right)dt<\infty .\right\}$$

Then, by [1,2,3], problem (1)–(4) has a unique solution, given by the optimal feedback control

$$u_{\ast }=u_{\ast }\left[x_{\ast }\left(t\right)\right]=-H^{-1}{B}^T{K}_{\ast }{x}_{\ast }\left(t\right),$$

(6)

where the symmetric matrix $K_{\ast }$ is the unique positive definite solution of the algebraic Riccati equation

$$A^{T}K+KA-KB{H}^{-1}{B}^{T}K+Q=0.$$

(7)

The optimal value of the objective functional is as follows:

$$J\left(u_{\ast }\right)={\left(x^0\right)}^T{K}_{\ast }{x}^0.$$

Henceforth, the optimal solution notation is simplified by omitting the superscript asterisk where context permits.

In [1] it was noted that this result remains true in the case of symmetric no-negative matrix Q. However, in the general case such a statement is incorrect. In [3] the existence and sufficient conditions for the validity of such a generalization are found.

In this work we focus attention on the critical case of the optimal stabilization problem—that is, the situation in which the spectrum of the system matrix contains purely imaginary eigenvalues or the standard positive-definiteness conditions on the weight matrices of the objective functional are violated. In this case some inclusions from (5) are violated.

3. Spectral Analysis of the Optimal Stabilization Problem

Let

$$G(x\left(t\right),u\left(t\right))=x^T\left(t\right)Qx\left(t\right)+u^T\left(t\right)Hu\left(t\right).$$

(8)

We extend the quadratic form $G(x,u)$ to the complex values of the vectors $x,u$ while preserving Hermitianity

$$G(\tilde{x},\tilde{u})={\tilde{x}}^{\star }Q\tilde{x}+{\tilde{u}}^{\star }H\tilde{u}.$$

(9)

Hereinafter, the asterisk ⋆ denotes Hermitian conjugation: transpose and complex conjugation.

In (9) $\tilde{x},\tilde{u}$ are arbitrary complex vectors. Let them satisfy the equation

$$\iota \omega \tilde{x}=A\tilde{x}+B\tilde{u},$$

where $\omega$ is an arbitrary real number, ${\iota }^2=-1$.

For the program control of the optimal control problem (2)–(4) the necessary optimality conditions in the form of the maximum principle Pontryagin hold. To formulate them, we write the Hamilton function

$$\mathcal{H}(x,u,\psi )=-G(x\left(t\right),u\left(t\right))+{\psi }^T(Ax+Bu).$$

Then, if the pair $(x=x(t),u=u(t\left)\right)$ is the solution of problem (2)–(4) in the class of functions $W_2^{1,n}(0,\infty )\times L_2^r(0,\infty ),$ then there exists a function $\psi \left(t\right)\in W_2^{1,n}(0,\infty ),$ such that pair $(x,\psi )$ satisfies the equations

$${\displaystyle \frac{dx}{dt}}={\left({\displaystyle \frac{\partial \mathcal{H}}{\partial \psi }}\right)}^T,{\displaystyle \frac{d\psi }{dt}}=-{\left({\displaystyle \frac{\partial \mathcal{H}}{\partial x}}\right)}^T,$$

and for optimal control we obtain

$${\displaystyle \frac{\partial \mathcal{H}}{\partial u}}=0\Rightarrow u=-{\displaystyle \frac{1}{2}}{H}^{-1}{B}^T\psi .$$

(10)

The system (10) can be written as a matrix equation

$$\widehat{J}{\displaystyle \frac{dz}{dt}}=\widehat{K}z,$$

(11)

where $z^T=(x,\psi ),$

$$\widehat{J}=\left(\begin{array}{cc}0& -E_n\\ E_n& 0\end{array}\right),\widehat{K}=\left(\begin{array}{cc}-2Q& A^T\\ A& {\displaystyle \frac{1}{2}}B{H}^{-1}{B}^T\end{array}\right).$$

Let the system (11) have no purely imaginary eigenvalues, i.e.,

$$\phi \left(\iota \omega \right)=det\left({\widehat{J}}^{-1}\widehat{K}-\iota \omega E_{2n}\right)\ne 0\mathrm{for} \mathrm{all}\omega \in R^1.$$

(12)

In [3] it was established that the condition (12) is equivalent to the condition

$$\begin{array}{c}G(\tilde{x},\tilde{u})>0\mathrm{for} \mathrm{all}\omega \in R^1\\ \mathrm{and} \mathrm{for} \mathrm{all}\tilde{x},\tilde{u}\mathrm{such} \mathrm{that}|\tilde{x}{|}^2+{|\tilde{u}|}^2\ne 0:\iota \omega \tilde{x}=A\tilde{x}+B\tilde{u}.\end{array}$$

(13)

In the case where the matrix A has no purely imaginary eigenvalues, condition (13) is equivalent to the condition

$$G\left({(\iota \omega E_n-A)}^{-1}B\tilde{u},\tilde{u}\right)>0,\forall \omega \in R^1,\forall \tilde{u}\ne 0.$$

(14)

From (14) follows the inequality

$$G(\tilde{x},\tilde{u})\ge {\tilde{u}}^{\star }H\tilde{u}>0,\forall \tilde{u}\ne 0.$$

Let the symmetric matrix $\Pi (\iota \omega$) of dimension $r\times r$ be defined by the equality

$$G\left({(\iota \omega E_n-A)}^{-1}B\tilde{u},\tilde{u}\right)={\tilde{u}}^{\star }\Pi \left(\iota \omega \right)\tilde{u}.$$

(15)

Then we have the representation

$$\phi \left(\iota \omega \right)=2^r{(detH)}^{-1}|\delta \left(\iota \omega \right){|}^2\Pi \left(\iota \omega \right),$$

(16)

where

$$\delta \left(\lambda \right)=det(\lambda E_n-A).$$

If the matrix A has purely imaginary eigenvalues, then the problem (2)–(4) may not have solutions in the class of functions $W_2^{1,n}(0,\infty )\times L_2^r(0,\infty ).$

Let all purely imaginary eigenvalues $\iota {\omega }_j$ of the matrix A be distinct. Let $p_j$ be corresponding eigenvectors, i.e.,

$$A{p}_j=\iota {\omega }_j{p}_j,p_j\ne 0.$$

Let us show that in the case of the form (8) the necessary and sufficient condition for the existence of optimal control is the fulfillment of the inequalities

$$p_j^{\star }Q{p}_j>0\mathrm{for} \mathrm{all}j.$$

(17)

Indeed, from (13) for $\omega ={\omega }_j,$ $\tilde{x}=p_j$ and $\tilde{u}=0$ follows (17).

Let us further assume that condition (17) is satisfied, but condition (13) is violated:

$$\exists {\omega }_0,{\tilde{x}}_0,{\tilde{u}}_0:\iota {\omega }_0{\tilde{x}}_0=A{\tilde{x}}_0+B{\tilde{u}}_0,|{\tilde{x}}_0{|}^2+{\left|{\tilde{u}}_0\right|}^2\ne 0,G({\tilde{x}}_0,{\tilde{u}}_0)=0.$$

Then $Q{\tilde{x}}_0=0$ and ${\tilde{u}}_0=0.$ Hence ${\omega }_0={\omega }_j$ and ${\tilde{x}}_0=p_j$ for some $j,$ so $G({\tilde{x}}_0,{\tilde{u}}_0)=p_j^{\star }Q{p}_j=0,$ which contradicts (17).

In the general case, taking into account the representation (16), the condition (13) in terms of $\Pi \left(\iota \omega \right)$, defined by (16), is equivalent to the inequalities

$$det>0,\underset{\omega \to {\omega }_j}{lim}|det\left(\iota \omega E_n-A\right){|}^{2}det\Pi \left(\iota \omega \right)>0.$$

(18)

Inequalities (18) follow from [3] (Theorem 2.1 on p. 73).

Thus, the necessary and sufficient conditions for the existence of a solution to the optimal control problem (2)–(4) in the case of a matrix A with an imaginary spectrum and a non-negative symmetric matrix Q are reduced to the fulfillment of all inequalities (17), or the second inequality from (18). The optimal stabilization problem (2)–(4) in the case of a matrix A with an imaginary spectrum has no a solution if the inequalities from (17) or (18) are violated, or

$$\phi \left(\iota {\omega }_j\right)=0$$

at least for one imaginary eigenvalue of the matrix $A.$

4. Optimal Stabilization in the Case of n = 2, r = 1

Consider the optimal stabilization problem with the following data:

$$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right),B=\left(\begin{array}{c}{b}_{11}\\ b_{21}\end{array}\right),Q=\left(\begin{array}{cc}{q}_{11}& q_{12}\\ q_{12}& q_{22}\end{array}\right),H=1.$$

The matrix A has purely imaginary eigenvalues; that is, the characteristic equation of A takes the form

$${\lambda }^2-(a_{11}+a_{22})\lambda +a_{11}{a}_{22}-a_{12}{a}_{21}=0,$$

and its coefficients satisfy the conditions

$$a_{11}+a_{22}=0,a_{11}^2+a_{12}{a}_{21}\le 0.$$

Let

$${\omega }_{1,2}=\pm \sqrt{|a_{11}^2+a_{12}{a}_{21}|}.$$

The pair $(A,B)$ is completely controllable, i.e.,

$$rank\left[B,AB\right]=2.$$

The matrix Q is symmetric and non-negative definite. Necessary and sufficient conditions for this are

$$q_{ii}\ge 0,i=\overline{1,2},q_{11}+q_{22}\ne 0,q_{11}{q}_{22}-q_{12}^2\ge 0.$$

Then

$$\widehat{J}=\left(\begin{array}{cccc}0& 0& \hfill -1& \hfill 0\\ 0& 0& \hfill 0& \hfill -1\\ 1& 0& \hfill 0& \hfill 0\\ 0& 1& \hfill 0& \hfill 0\end{array}\right),\widehat{K}=\left(\begin{array}{cccc}\hfill -2q_{11}& \hfill -2q_{12}& a_{11}& a_{21}\\ \hfill -2q_{12}& \hfill -2q_{22}& a_{12}& a_{22}\\ \hfill a_{11}& \hfill a_{12}& {\displaystyle \frac{1}{2}}{b}_{1,1}^2& {\displaystyle \frac{1}{2}}{b}_{11}{b}_{21}\\ \hfill a_{21}& \hfill a_{22}& {\displaystyle \frac{1}{2}}{b}_{11}{b}_{21}& {\displaystyle \frac{1}{2}}{b}_{21}^2\end{array}\right),$$

$$\phi \left(\iota \omega \right)=det\left({\widehat{J}}^{-1}\widehat{K}-\iota \omega E_4\right).$$

For simplicity of exposition, we set

$$a_{11}=a_{22}=0,a_{12}=1,a_{21}=-\alpha ,\alpha \ge 0;q_{11}=q_{12}=0,q_{22}=\beta >0;H=1.$$

The above conditions on the matrices A, B, Q are thus satisfied, confirming that this is a critical case of the optimal stabilization problem. In the frequency domain, system (2) corresponds to

$$\iota \omega {\tilde{x}}_1={\tilde{x}}_2,\iota \omega {\tilde{x}}_2=-\alpha {\tilde{x}}_1+\tilde{u},$$

whose solution is given by

$${\tilde{x}}_1={\left(\alpha -{\omega }^2\right)}^{-1}\tilde{u},{\tilde{x}}_2=\iota \omega {\left(\alpha -{\omega }^2\right)}^{-1}\tilde{u}.$$

Since $G(\tilde{x},\tilde{u})=\beta |{\tilde{x}}_2{|}^2+|\tilde{u}{|}^2=\Pi \left(\iota \omega \right){\left|\tilde{u}\right|}^2$ we obtain

$$\Pi \left(\iota \omega \right)=1+\beta {\omega }^2{\left(\alpha -{\omega }^2\right)}^{-2}.$$

Since $\delta \left(\lambda \right)={\lambda }^2+\alpha$, it follows that

$$\phi \left(\iota \omega \right)={\Pi \left(\iota \omega \right)\left|\delta \left(\iota \omega \right)\right|}^2={\left(\alpha -{\omega }^2\right)}^2+\beta {\omega }^2.$$

The condition $\alpha >0$ ensures that

$$\phi \left(\iota \omega \right)>0\forall \omega \in R^1,$$

thereby guaranteeing the existence of a solution to the optimal stabilization problem.

In this case, the algebraic Riccati system (7) reduces to

$$\begin{array}{c}{K}_{12}^2+2\alpha K_{12}=0,\\ K_{11}-\alpha K_{22}-K_{12}{K}_{22}=0,\\ 2K_{1,2}-K_{2,2}^2+\beta =0.\end{array}$$

Its unique positive definite solution is

$$K=\left(\begin{array}{cc}\alpha \sqrt{\beta }& 0\\ 0& \sqrt{\beta }\end{array}\right).$$

The optimal feedback control is

$$u=-H^{-1}{B}^{T}Kx=-\sqrt{\beta }{x}_2.$$

(19)

The corresponding closed-loop system is

$${\dot{x}}_1=x_2,{\dot{x}}_2=-\alpha x_1-\sqrt{\beta }{x}_2.$$

(20)

The characteristic equation of this system is

$${\lambda }^2+\sqrt{\beta }\lambda +\alpha =0$$

with eigenvalues

$${\lambda }_{1,2}=-{\displaystyle \frac{\sqrt{\beta }}{2}}\pm \sqrt{{\displaystyle \frac{\beta }{4}}-\alpha }.$$

Depending on the relationship between the parameters $\alpha$ and $\beta$, three cases for the eigenvalues are possible; however, all of them have strictly negative real parts. For definiteness, consider the case of distinct real eigenvalues, i.e., assume ${\displaystyle \frac{\beta }{4}}-\alpha >0.$ Then

$$\begin{array}{c}{x}_1\left(t\right)={\displaystyle \frac{x_2^0-{\lambda }_2{x}_1^0}{{\lambda }_1-{\lambda }_2}}exp\left({\lambda }_{1}t\right)-{\displaystyle \frac{x_2^0-{\lambda }_1{x}_1^0}{{\lambda }_1-{\lambda }_2}}exp\left({\lambda }_{2}t\right),\\ x_2\left(t\right)={\lambda }_2{\displaystyle \frac{x_2^0-{\lambda }_2{x}_1^0}{{\lambda }_1-{\lambda }_2}}exp\left({\lambda }_{1}t\right)-{\lambda }_2{\displaystyle \frac{x_2^0-{\lambda }_1{x}_1^0}{{\lambda }_1-{\lambda }_2}}exp\left({\lambda }_{2}t\right),{\lambda }_2>{\lambda }_1.\end{array}$$

Now let $\alpha =0$. Then for $\omega =0$ we have

$$\phi \left(\iota \omega \right)=0.$$

We verify condition (18) for the existence of a solution to the optimal stabilization problem:

$$\underset{\omega \to {\omega }_j}{lim}|det\left(\iota \omega E_n-A\right){|}^{2}det\Pi \left(\iota \omega \right)=\underset{\omega \to 0}{lim}{\omega }^4(1+\beta {\omega }^{-2})=0.$$

Hence, the optimal stabilization problem has no solution in this case.

Having an explicit analytical solution for $\alpha >0$, we now examine the qualitative transition to the limiting case $\alpha =0$.

The matrix K then has the form

$$K=\left(\begin{array}{cc}0& 0\\ 0& \sqrt{\beta }\end{array}\right).$$

The control is again given by (19), and the corresponding solution of system (20) is

$$x_1\left(t\right)=x_1^0+{\displaystyle \frac{x_2^0}{\sqrt{\beta }}}(1-exp(-\sqrt{\beta }t)),x_2\left(t\right)=x_2^{0}exp(-\sqrt{\beta }t).$$

(21)

From the above example, the following conclusions can be drawn:

• The problem (1)–(4) with matrices

  $$A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),B=\left(\begin{array}{c}0\\ 1\end{array}\right),Q=\left(\begin{array}{cc}0& 0\\ 0& \beta \end{array}\right),H=1$$

  admits no solution in the space $\mathrm{Y}(x,u)$ for $n=2$, $r=1$; however, the solution constructed above exists and belongs to the spaces $u\left(t\right)\in L_2(0,\infty ),$ $x_2\left(t\right)\in W_2^{1,1}(0,\infty ),$ ${\dot{x}}_1\left(t\right)\in L_2(0,\infty ),$ $x_1\left(t\right)\in C[0,\infty ),$ where $C[0,\infty )$ denotes the space of continuous functions.
• For arbitrary $\xi ,x_2^0\in R^1$ there exists a number $x_1^0=\xi -{\displaystyle \frac{x_2^0}{\sqrt{\beta }}}$ such that

  $$\underset{t\to \infty }{lim}{x}_1\left(t\right)=\xi ,\underset{t\to \infty }{lim}{x}_2\left(t\right)=0,$$

  in particular, for $x_1^0+{\displaystyle \frac{x_2^0}{\sqrt{\beta }}}=0$ we have $\xi =0.$

5. Regularization of the Optimal Stabilization Problem in the Critical Case

Thus, the optimal stabilization problem (1)–(4) with a symmetric non-negative matrix Q and a matrix A whose spectrum contains no purely imaginary eigenvalues always possesses a unique solution in the form of the optimal regulator (6), and the inclusion (5) holds. If the spectrum of A is purely imaginary, then a unique solution still exists provided that either condition (17) or the boundary inequalities (18) are additionally satisfied. Otherwise, problem (1)–(4) has no solution in the class (5); this is the critical case.

Denote by ${\mathrm{Y}}_1(x,u)$ the space of pairs $\left(x\left(t\right),u\left(t\right)=u\left[x\left(t\right)\right]\in L_2(0,\infty )\right)$ for which $J\left(u\right)<\infty$. By strict convexity of the functional, the problem admits a unique solution; however, this solution does not belong to the space $\mathrm{Y}(x,u)$.

Two approaches are available for treating the critical case:

• Regard it as degenerate, in the sense that no solution to the optimal stabilization problem exists in the space $\mathrm{Y}(x,u)$ for all $x^0\in R^n$;
• Identify additional conditions on the data of problem (1)–(4) under which a solution exists in $\mathrm{Y}(x,u)$ (or in ${\mathrm{Y}}_1(x,u)$).

We first consider the former approach. The critical problem (1)–(4) may be regularized in two ways:

• By perturbing the weight matrix Q of the objective functional;
• By perturbing the system matrix A.

5.1. Regularization by Perturbation of the Matrix Q

We replace problem (1)–(4) with the following regularized problem: for a sufficiently small $\epsilon \in (0,\widehat{\epsilon }]$, find a feedback control

$$u^{\epsilon }=u^{\epsilon }\left[x^{\epsilon }\left(t\right)\right],$$

(22)

where the function $x^{\epsilon }\left(t\right)$ describes the controlled process and satisfies the Cauchy problem

$$\dot{x^{\epsilon }}\left(t\right)=A{x}^{\epsilon }\left(t\right)+B{u}^{\epsilon }\left[x^{\epsilon }\left(t\right)\right],0<t\le \infty ,$$

(23)

$$x^{\epsilon }\left(0\right)=x^0,$$

(24)

such that the objective functional

$$J^{\epsilon }\left(u^{\epsilon }\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left({\left(x^{\epsilon }\left(t\right)\right)}^T{Q}^{\epsilon }{x}^{\epsilon }\left(t\right)+{\left(u^{\epsilon }\left(t\right)\right)}^{T}H{u}^{\epsilon }\left(t\right)\right)dt$$

(25)

is minimized and the following approximation estimates hold:

$$\begin{array}{c}|J^{\epsilon }\left(u^{\epsilon }\right)-J\left(u\right)|\le C_1\epsilon ,\\ \left|\right|u^{\epsilon }\left(t\right)-u\left(t\right){\left|\right|}_{L_2(0,\infty )}\le C_2\epsilon .\end{array}$$

(26)

Here $C_i$, $i=\overline{1,2}$, are positive constants independent of $\epsilon$.

In problem (22)–(25), the matrix $Q^{\epsilon }$ is symmetric, positive definite, and smooth in $\epsilon$, and satisfies $Q^0=Q$. The optimal controller then takes the form

$$u^{\epsilon }=u^{\epsilon }\left[x^{\epsilon }\left(t\right)\right]=-H^{-1}{B}^T{K}^{\epsilon }{x}^{\epsilon }\left(t\right),$$

where $K^{\epsilon }$ is the unique positive definite solution of the algebraic Riccati equation

$$A^T{K}^{\epsilon }+K^{\epsilon }A-K^{\epsilon }B{H}^{-1}{B}^T{K}^{\epsilon }+Q^{\epsilon }=0.$$

(27)

The optimal value of the objective functional is

$$J^{\epsilon }\left(u^{\epsilon }\right)={\left(x^0\right)}^T{K}^{\epsilon }{x}^0.$$

We now establish the first estimate in (26).

$$|J^{\epsilon }\left(u^{\epsilon }\right)-J\left(u\right)|=|{\left(x^0\right)}^T(K^{\epsilon }-K)x^0| \le \parallel K^{\epsilon }-K\parallel \parallel x^0{\parallel }^2,$$

where the matrix and vector norms are defined by

$$\parallel W\parallel =\underset{i,j=\overline{1,n}}{max}|w_{i,j}|,\parallel v\parallel =\underset{i=\overline{1,n}}{max}\left|v_i\right|.$$

When solving Equation (27) directly is difficult, an approximate solution may be sought in the form of the power series

$$K^{\epsilon }={\displaystyle \sum _{i=0}^{\infty }}{\widehat{K}}_i{\epsilon }^i,$$

where the matrices ${\widehat{K}}_i$ satisfy the linear matrix equations

$$A^T{\widehat{K}}_i+{\widehat{K}}_{i}A-{\displaystyle \sum _{j=0}^i}{\widehat{K}}_{j}B{H}^{-1}{B}^T{\widehat{K}}_{i-j}+{\displaystyle \frac{1}{i!}}{\left.{\displaystyle \frac{d^i{Q}^{\epsilon }}{d{\epsilon }^i}}\right|}_{\epsilon =0}=0.$$

For example from Section 4 let us set $Q^{\epsilon }=Q+{\epsilon }^2{E}_2,\epsilon \in (0,\widehat{\epsilon }].$ Here, the squared parameter ${\epsilon }^2$ is specifically chosen instead of $\epsilon$ to ensure that the perturbing term remains strictly positive for any real $\epsilon \ne 0$, thereby guaranteeing the strict positive definiteness of $Q^{\epsilon }$. Then the system (27) takes the form

$$\begin{array}{c}{\left(K_{12}^{\epsilon }\right)}^2-{\epsilon }^2=0,\\ K_{11}^{\epsilon }-K_{12}^{\epsilon }{K}_{22}^{\epsilon }=0,\\ 2K_{12}^{\epsilon }-{\left(K_{22}^{\epsilon }\right)}^2+\beta +{\epsilon }^2=0.\end{array}$$

(28)

The system (28) has a unique positive definite solution

$$K^{\epsilon }=\left(\begin{array}{cc}\epsilon \sqrt{\beta +2\epsilon +{\epsilon }^2}& \epsilon \\ \epsilon & \sqrt{\beta +2\epsilon +{\epsilon }^2}\end{array}\right).$$

Then

$$K^{\epsilon }-K=\epsilon \left(\begin{array}{cc}\sqrt{\beta +2\epsilon +{\epsilon }^2}& 1\\ 1& {\displaystyle \frac{\epsilon +2}{\sqrt{\beta +2\epsilon +{\epsilon }^2}+\sqrt{\beta }}}\end{array}\right),$$

and therefore

$$\begin{array}{c}\parallel K^{\epsilon }-K\parallel \le \epsilon max\left\{\sqrt{\beta +2\widehat{\epsilon }+{\widehat{\epsilon }}^2},1,{\displaystyle \frac{\widehat{\epsilon }+2}{2\sqrt{\beta }}}\right\},\\ |J_{\epsilon }\left(u_{\epsilon }\right)-J\left(u\right)|\le C_1\epsilon ,\end{array}$$

where $C_1=max\left\{\sqrt{\beta +2\widehat{\epsilon }+{\widehat{\epsilon }}^2},1,{\displaystyle \frac{\widehat{\epsilon }+2}{2\sqrt{\beta }}}\right\}{\parallel x^0\parallel }^2.$

The regularized optimal control takes the form

$$u^{\epsilon }=-\epsilon x_1^{\epsilon }\left(t\right)-\sqrt{\beta +2\epsilon +{\epsilon }^2}{x}_2^{\epsilon }\left(t\right),$$

(29)

where $x^{\epsilon }\left(t\right)$ is the solution of the closed-loop regularized system

$$\begin{array}{c}{\dot{x}}_1^{\epsilon }\left(t\right)=x_2^{\epsilon }\left(t\right),\\ {\dot{x}}_2^{\epsilon }\left(t\right)=-\epsilon x_1^{\epsilon }\left(t\right)-\sqrt{\beta +2\epsilon +{\epsilon }^2}{x}_2^{\epsilon }\left(t\right),\\ x_1^{\epsilon }\left(0\right)=x_1^0,x_2^{\epsilon }\left(0\right)=x_2^0.\end{array}$$

(30)

The characteristic equation corresponding to the system (30) is

$${\left({\lambda }^{\epsilon }\right)}^2+\sqrt{\beta +2\epsilon +{\epsilon }^2}{\lambda }^{\epsilon }+\epsilon =0,$$

which has for sufficiently small $\epsilon$ two real negative roots

$${\lambda }_{1,2}^{\epsilon }=-{\displaystyle \frac{\sqrt{\beta +2\epsilon +{\epsilon }^2}}{2}}\mp {\displaystyle \frac{\sqrt{\beta -2\epsilon +{\epsilon }^2}}{2}},{\lambda }_1^{\epsilon }<{\lambda }_2^{\epsilon }.$$

The solution to the Cauchy problem (30) is

$$\begin{array}{c}{x}_1^{\epsilon }\left(t\right)=-{\displaystyle \frac{x_2^0-{\lambda }_2^{\epsilon }{x}_1^0}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_1^{\epsilon }t\right)+{\displaystyle \frac{x_2^0-{\lambda }_1^{\epsilon }{x}_1^0}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_2^{\epsilon }t\right),\\ x_2^{\epsilon }\left(t\right)=-{\displaystyle \frac{{\lambda }_1^{\epsilon }\left(x_2^0-{\lambda }_2^{\epsilon }{x}_1^0\right)}{{\lambda }_2^{\epsilon }-{\lambda }_{\epsilon }^1}}exp\left({\lambda }_1^{\epsilon }t\right)+{\displaystyle \frac{{\lambda }_2^{\epsilon }\left(x_2^0-{\lambda }_1^{\epsilon }{x}_1^0\right)}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_2^{\epsilon }t\right).\end{array}$$

(31)

It is obvious that

$$\underset{t\to \infty }{lim}{x}_i^{\epsilon }\left(t\right)=0,i=\overline{1,2},\epsilon >0.$$

In this example, it is possible to investigate the limit transition of the solution (31) to the solution of the critical system (21) for $\epsilon \to 0.$

First,

$$\underset{\epsilon \to 0}{lim}{\lambda }_1^{\epsilon }=-\sqrt{\beta },\underset{\epsilon \to 0}{lim}{\lambda }_2^{\epsilon }=0.$$

Secondly,

$$\underset{\epsilon \to 0}{lim}{x}_2^{\epsilon }\left(t\right)=-\underset{\epsilon \to 0}{lim}{\displaystyle \frac{{\lambda }_1^{\epsilon }\left(x_2^0-{\lambda }_2^{\epsilon }{x}_1^0\right)}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_1^{\epsilon }t\right)+\underset{\epsilon \to 0}{lim}{\displaystyle \frac{{\lambda }_2^{\epsilon }\left(x_2^0-{\lambda }_1^{\epsilon }{x}_1^0\right)}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_2^{\epsilon }t\right)=$$

$$=x_2^{0}exp(-\sqrt{\beta }t),$$

$$\underset{\epsilon \to 0}{lim}{x}_1^{\epsilon }\left(t\right)=-\underset{\epsilon \to 0}{lim}{\displaystyle \frac{x_2^0-{\lambda }_2^{\epsilon }{x}_1^0}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_1^{\epsilon }t\right)+\underset{\epsilon \to 0}{lim}{\displaystyle \frac{x_2^0-{\lambda }_1^{\epsilon }{x}_1^0}{{\lambda }_2^{\epsilon }-{\lambda }_1^{\epsilon }}}exp\left({\lambda }_2^{\epsilon }t\right)=$$

$$=-x_2^0{\displaystyle \frac{exp\left(-\sqrt{\beta }t\right)}{\sqrt{\beta }}}+{\displaystyle \frac{x_2^0+\sqrt{\beta }{x}_1^0}{\sqrt{\beta }}}=x_1^0+{\displaystyle \frac{x_2^0}{\sqrt{\beta }}}\left(1-exp\left(-\sqrt{\beta }\right)t\right).$$

From (19) and (29) we have

$$\begin{array}{c}{u}^{\epsilon }\left[x^{\epsilon }\left(t\right)\right]-u\left[x\left(t\right)\right]=-\epsilon x_1^{\epsilon }\left(t\right)-\sqrt{\beta +2\epsilon +{\epsilon }^2}{x}_2^{\epsilon }\left(t\right)+\sqrt{\beta }{x}_2\left(t\right)\\ =-\epsilon x_1^{\epsilon }\left(t\right)-\left(\sqrt{\beta +2\epsilon +{\epsilon }^2}-\sqrt{\beta }\right)x_2^{\epsilon }\left(t\right)-\sqrt{\beta }{y}_2^{\epsilon }\left(t\right)\end{array}$$

(32)

where deviation $y_i^{\epsilon }\left(t\right)=x_i^{\epsilon }\left(t\right)-x_i\left(t\right),i=\overline{1,2}$ is a solution to the system of differential equations

$$\begin{array}{c}{\dot{y}}_1^{\epsilon }\left(t\right)=y_{2,\epsilon }\left(t\right),\\ {\dot{y}}_2^{\epsilon }\left(t\right)=-\epsilon x_1^{\epsilon }\left(t\right)-\left(\sqrt{\beta +2\epsilon +{\epsilon }^2}-\sqrt{\beta }\right)x_2^{\epsilon }\left(t\right)-\sqrt{\beta }{y}_2^{\epsilon }\left(t\right),\\ y_i^{\epsilon }\left(0\right)=0,i=\overline{1,2}.\end{array}$$

(33)

From the second equation of the system (33) we can write that

$$y_2^{\epsilon }\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}}exp\left(-\sqrt{\beta }(t-\tau )\right)f^{\epsilon }\left(\tau \right)d\tau ,$$

(34)

where

$$f^{\epsilon }\left(t\right)=-\epsilon x_1^{\epsilon }\left(t\right)-{\displaystyle \frac{2\epsilon +{\epsilon }^2}{\sqrt{\beta +2\epsilon +{\epsilon }^2}+\sqrt{\beta }}}{x}_2^{\epsilon }\left(t\right).$$

The inclusion $f^{\epsilon }\left(t\right)\in L_2(0,\infty )$ is true and the following inequality holds:

$$\parallel f^{\epsilon }{(·)\parallel }_{L_2(0,\infty )}\le C\epsilon .$$

Really,

$$\parallel f^{\epsilon }{(·)\parallel }_{L_2(0,\infty )}\le \epsilon \parallel x_1^{\epsilon }{(·)\parallel }_{L_2(0,\infty )}+{\displaystyle \frac{2\epsilon +{\epsilon }^2}{\sqrt{\beta +2\epsilon +{\epsilon }^2}+\sqrt{\beta }}}{\parallel x_2^{\epsilon }(·)\parallel }_{L_2(0,\infty )}\le C\epsilon .$$

Further, using the representation (34) and Cauchy’s inequality, we have

$$\begin{array}{c}{\left(y_2^{\epsilon }\left(t\right)\right)}^2\le {\displaystyle \underset{0}{\overset{t}{\int }}}exp\left(-2\sqrt{\beta }(t-\tau )\right)d\tau {\displaystyle \underset{0}{\overset{t}{\int }}}{\left(f^{\epsilon }\left(\tau \right)\right)}^{2}d\tau \le \\ \le {\displaystyle \underset{0}{\overset{t}{\int }}}exp(-2\sqrt{\beta }(t-\tau ))d\tau \parallel f^{\epsilon }{(·)\parallel }_{L_2(0,\infty )}^2.\end{array}$$

Hence, taking into account formulas (34), we obtain that

$$\parallel y_2^{\epsilon }{(·)\parallel }_{L_2(0,\infty )}\le C\epsilon .$$

So from the representation (32) we finally have the estimate

$$\parallel u^{\epsilon }\left[x^{\epsilon }\left(t\right)\right]-u\left[x\left(t\right)\right]{\left|\right|}_{L_2(0,\infty )}\le C\epsilon .$$

(35)

As for the deviation $y_1^{\epsilon }\left(t\right),$ it does not belong to the space $L_2(0,\infty ),$ but the limit equality take place

$$\underset{t\to \infty }{lim}{y}_1^{\epsilon }\left(t\right)=x_1^0+{\displaystyle \frac{x_2^0}{\sqrt{\beta }}},\epsilon >0.$$

(36)

Let ${\mathcal{A}}^{\epsilon }=\left(J^{\epsilon },u^{\epsilon }\left(t\right),x^{\epsilon }\left(t\right)\right)$ be the solution of the regularized (perturbed) problem of optimal stabilization (22)–(25), and ${\mathcal{A}}^0=\left(J,u\left(t\right),x\left(t\right)\right)$ be the solution of the problem of optimal stabilization in the critical case (degenerate). Then, according to the terminology of the work [19] and the equality (36), the regularized problem is singularly perturbed. Another thing is that here we do not need to look for a uniform in $\epsilon$ approximation $x_1^{\epsilon }\left(t\right)$ to $x_1\left(t\right).$

5.2. Regularization by Perturbation of the Matrix A

Let us replace the problem (1)–(4) with the following problem: for a sufficiently small real number $\epsilon \in (0,\widehat{\epsilon }]$ find

$$u^{\epsilon }=u^{\epsilon }\left[x^{\epsilon }\left(t\right)\right],$$

(37)

where the function $x^{\epsilon }\left(t\right)$ describes the controlled process and satisfies the Cauchy problem

$${\dot{x}}^{\epsilon }\left(t\right)=A^{\epsilon }{x}^{\epsilon }\left(t\right)+B{u}^{\epsilon }\left[x^{\epsilon }\left(t\right)\right],0<t<\infty ,$$

(38)

$$x^{\epsilon }\left(0\right)=x^0,$$

(39)

such that the objective functional

$$J^{\epsilon }\left(u^{\epsilon }\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}{\left(x^{\epsilon }\left(t\right)\right)}^{T}Q{x}^{\epsilon }\left(t\right)+{\left(u^{\epsilon }\left(t\right)\right)}^{T}H{u}^{\epsilon }\left(t\right))dt$$

(40)

would take the smallest value and the inequalities (26) would hold.

In problem (37)–(40) the matrix $A^{\epsilon }$ is smooth on the parameter $\epsilon$, such that $A^0=A$ and its spectrum is not imaginary. Then in problem (37)–(40) the optimal controller takes the form

$$u^{\epsilon }=u^{\epsilon }\left[x^{\epsilon }\left(t\right)\right]=-H^{-1}{B}^T{K}^{\epsilon }{x}^{\epsilon }\left(t\right),$$

where the matrix $K^{\epsilon }$ is the only positive definite solution of the algebraic Riccati equation

$${\left(A^{\epsilon }\right)}^T{K}^{\epsilon }+K^{\epsilon }{A}^{\epsilon }-K^{\epsilon }B{H}^{-1}{B}^T{K}^{\epsilon }+Q=0.$$

(41)

The optimal value of the objective functional is appearance

$$J^{\epsilon }\left(u^{\epsilon }\right)={\left(x^0\right)}^T{K}^{\epsilon }{x}^0.$$

(42)

Let us return to the example of Section 4 and put $A^{\epsilon }=A+\epsilon E_2,\epsilon \in (0,\widehat{\epsilon }].$ Then the system (41) takes the form

$$\begin{array}{c}2\epsilon K_{11}^{\epsilon }-{\left(K_{12}^{\epsilon }\right)}^2=0,\\ K_{11}^{\epsilon }+2\epsilon K_{12}^{\epsilon }-K_{12}^{\epsilon }{K}_{22}^{\epsilon }=0,\\ 2K_{12}^{\epsilon }+2\epsilon K_{22}^{\epsilon }-{\left(K_{22}^{\epsilon }\right)}^2+\beta =0.\end{array}$$

(43)

The system (43) has a unique positive definite solution

$$K^{\epsilon }=\left(\begin{array}{cc}2\epsilon {\left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)}^2& 2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)\\ 2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)& 3\epsilon +\sqrt{\beta +{\epsilon }^2}\end{array}\right).$$

So,

$$\begin{array}{c}{K}^{\epsilon }-K=\epsilon \left(\begin{array}{cc}2{\left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)}^2& 2\left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)\\ 2\left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)& 3+{\displaystyle \frac{\epsilon }{\sqrt{\beta +{\epsilon }^2}+\sqrt{\beta }}}\end{array}\right),\\ \parallel K^{\epsilon }-K\parallel \le \epsilon max\left\{2{\left(\widehat{\epsilon }+\sqrt{\beta +{\widehat{\epsilon }}^2}\right)}^2,2\left(\widehat{\epsilon }+\sqrt{\beta +{\widehat{\epsilon }}^2}\right),3+{\displaystyle \frac{\widehat{\epsilon }}{2\sqrt{\beta }}}\right\}\\ |J^{\epsilon }\left(u^{\epsilon }\right)-J\left(u\right)|\le C_1\epsilon ,\end{array}$$

where

$$C_{1}max\left\{2{\left(\widehat{\epsilon }+\sqrt{\beta +{\widehat{\epsilon }}^2}\right)}^2,2\left(\widehat{\epsilon }+\sqrt{\beta +{\widehat{\epsilon }}^2}\right),3+{\displaystyle \frac{\widehat{\epsilon }}{2\sqrt{\beta }}}\right\}{\parallel x^0\parallel }^2.$$

The regularized optimal control is

$$u_{\epsilon }=-2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_1^{\epsilon }\left(t\right)-\left(3\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_2^{\epsilon }\left(t\right),$$

(44)

where $x_{\epsilon }\left(t\right)$ is the solution of the closed-loop regularized system

$$\begin{array}{c}{\dot{x}}_1^{\epsilon }\left(t\right)=\epsilon x_1^{\epsilon }\left(t\right)+x_2^{\epsilon }\left(t\right),\\ {\dot{x}}_2^{\epsilon }\left(t\right)=-2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_1^{\epsilon }\left(t\right)-\left(2\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_2^{\epsilon }\left(t\right),\\ x_1^{\epsilon }\left(0\right)=x_1^0,x_2^{\epsilon }\left(0\right)=x_2^0.\end{array}$$

(45)

The characteristic equation corresponding to the system (45) is

$${\left({\lambda }^{\epsilon }\right)}^2+\left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right){\lambda }^{\epsilon }+\epsilon \sqrt{\beta +{\epsilon }^2}=0,$$

which has two real negative roots

$${\lambda }_1^{\epsilon }=-\sqrt{\beta +{\epsilon }^2},{\lambda }_2^{\epsilon }=-\epsilon ,{\lambda }_1^{\epsilon }<{\lambda }_2^{\epsilon }.$$

From (19) and (44) we have

$$\begin{array}{c}{u}^{\epsilon }\left[x^{\epsilon }\left(t\right)\right]-u\left[x\left(t\right)\right]=-2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_1^{\epsilon }\left(t\right)-\\ -\left(3\epsilon +\sqrt{\beta +{\epsilon }^2}-\sqrt{\beta }\right)x_2^{\epsilon }\left(t\right)-\sqrt{\beta }{y}_2^{\epsilon }\left(t\right),\end{array}$$

where deviations $y_i^{\epsilon }\left(t\right)=x_i^{\epsilon }\left(t\right)-x_i\left(t\right),i=\overline{1,2}$ are solutions of the system of differential equations

$$\begin{array}{c}{\dot{y}}_1^{\epsilon }\left(t\right)=\epsilon x_{1,\epsilon }\left(t\right)+y_2^{\epsilon }\left(t\right),\\ {\dot{y}}_2^{\epsilon }\left(t\right)=-2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_1^{\epsilon }\left(t\right)-\left(2\epsilon +\sqrt{\beta +{\epsilon }^2}-\sqrt{\beta }\right)x_2^{\epsilon }\left(t\right)-\sqrt{\beta }{y}_2^{\epsilon }\left(t\right),\\ y_i^{\epsilon }\left(0\right)=0,i=\overline{1,2}.\end{array}$$

The estimate $\parallel u^{\epsilon }\left[x^{\epsilon }(·)\right]-u\left[x(·)\right]{\parallel }_{L_2(0,\infty )}$ is established here similarly to the corresponding estimate for the case of regularization by perturbation of the matrix $Q.$

However, in the case of regularization by perturbation of the matrix A, we are more interested in the estimate $\parallel u^{\epsilon }\left[x^{\epsilon }(·)\right]-u\left[x(·)\right]{\parallel }_{L_2(0,\infty )},$ when the vector $x^{\epsilon }\left(t\right)$ is the solution of the closed-loop system generated by the critical system and the regularized control (44), i.e.,

$$\begin{array}{c}{\dot{x}}_1^{\epsilon }\left(t\right)=x_{2,\epsilon }\left(t\right),\\ {\dot{x}}_2^{\epsilon }\left(t\right)=-2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_1^{\epsilon }\left(t\right)-\left(3\epsilon +\sqrt{\beta +{\epsilon }^2}\right)x_2^{\epsilon }\left(t\right),\\ x_1^{\epsilon }\left(0\right)=x_1^0,x_2^{\epsilon }\left(0\right)=x_2^0.\end{array}$$

(46)

The characteristic equation corresponding to the system (46) is

$${\left({\lambda }^{\epsilon }\right)}^2+\left(3\epsilon +\sqrt{\beta +{\epsilon }^2}\right){\lambda }^{\epsilon }+2\epsilon \left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right)=0,$$

which has two real negative roots

$${\lambda }_1^{\epsilon }=-\left(\epsilon +\sqrt{\beta +{\epsilon }^2}\right),{\lambda }_2^{\epsilon }=-2\epsilon ,{\lambda }_1^{\epsilon }<{\lambda }_2^{\epsilon }.$$

Further, all calculations that were characteristic of the regularization of the functional are repeated. As a result, we obtain the estimate (35), and from it the estimate

$$|J\left(u^{\epsilon }\right)-J\left(u\right)| \le C\epsilon .$$

6. Special System of Optimal Stabilization in the Critical Case

We now consider, in the general setting, the possibilities for solving the optimal stabilization problem in the critical case (cf. Section 4).

Since the general problem is quite complex, a sequence of problems of increasing complexity is investigated before considering the most general case. First, in Section 6.2, the so-called canonical problem—where the system matrix is an n-dimensional nilpotent Jordan block and the control is scalar—is considered. In doing so, the cases where the objective functional involves one, two, and i coordinates are consistently studied. Then, in Section 6.2, the general case with scalar control is presented, and finally, in Section 6.3, the case with multidimensional control is investigated. Lastly, an illustrative example is considered to demonstrate the practical application of the algorithm.

6.1. Solving the Optimal Stabilization Problem for a Canonical System

We begin by considering the optimal stabilization problem for the canonical system

$$\begin{array}{c}{\dot{x}}_1\left(t\right)=x_2\left(t\right),\\ {\dot{x}}_2\left(t\right)=x_3\left(t\right),\\ \dots \dots \dots \dots \dots \\ {\dot{x}}_{n-1}\left(t\right)=x_n\left(t\right),\\ {\dot{x}}_n\left(t\right)=u\left(t\right),\\ x_i\left(0\right)=x_i^0,i=\overline{1,n},\end{array}$$

(47)

with the objective functional

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(x^T\left(t\right)Qx\left(t\right)+u^2\left(t\right)\right)dt,$$

(48)

where the matrix Q is symmetric and non-negative and has dimensions $n\times n.$

In [20], system (47) serves as the basis for constructing algorithms for constrained nonlinear synthesis via the controllability function method; complete controllability of the canonical system is also established therein. Throughout this section we assume that the optimal stabilization problem (47) and (48) is critical.

6.1.1. The “Weakest” Optimal Stabilization Is One-Coordinate

Suppose the matrix Q has the following structure:

$$q_{ij}=0,i=\overline{1,n-1},j=\overline{1,n};q_{nj}=0,j=\overline{1,n-1};q_{nn}>0.$$

Then the coordinate $x_n\left(t\right)$ satisfies a one-dimensional optimal stabilization problem: find the control $u=u\left[x_n\left(t\right)\right]$ that minimizes the functional

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(q_{nn}{x}_n^2\left(t\right)+u^2\left(t\right)\right)dt$$

subject to the Cauchy problem

$${\dot{x}}_n\left(t\right)=u\left(t\right),x_n\left(0\right)=x_n^0.$$

(49)

The optimal control is given by

$$u=-\sqrt{q_{nn}}{x}_n\left(t\right).$$

The corresponding solution of the Cauchy problem (49) is

$$x_n\left(t\right)=x_n^{0}exp\left(-\sqrt{q_{nn}}t\right)$$

and the optimal value of the objective functional is

$$J\left(u\right)=\sqrt{q_{nn}}{\left(x_n^0\right)}^2.$$

In this case, the following regularity inclusions hold: $x_n\left(t\right)\in L_2(0,\infty ),$ $u\left(t\right)\in L_2(0,\infty ).$

By the canonical structure of system (47), the preceding two coordinates are

$$\begin{array}{c}{x}_{n-1}\left(t\right)=x_{n-1}^0+{\displaystyle \frac{x_n^0}{\sqrt{q_{nn}}}}\left(1-exp\left(-\sqrt{q_{nn}}t\right)\right),\\ x_{n-2}\left(t\right)=x_{n-2}^0+\left(x_{n-1}^0+{\displaystyle \frac{x_n^0}{\sqrt{q_{nn}}}}\right)t-{\displaystyle \frac{x_n^0}{q_{nn}}}\left(1-exp\left(-\sqrt{q_{nn}}t\right)\right).\end{array}$$

(50)

Here $x_{n-1}\left(t\right)\in C[0,\infty )$, whereas $x_{n-2}\left(t\right)\notin C[0,\infty )$ in general. For $x_{n-2}\left(t\right)$ to belong to $C[0,\infty )$, it is necessary and sufficient that

$$x_{n-1}^0+{\displaystyle \frac{x_n^0}{\sqrt{q_{nn}}}}=0.$$

Under this condition, the coordinates (50) reduce to

$$\begin{array}{c}{x}_{n-1}\left(t\right)=-{\displaystyle \frac{x_n^0}{\sqrt{q_{nn}}}}exp\left(-\sqrt{q_{nn}}t\right),\\ x_{n-2}\left(t\right)=x_{n-2}^0-{\displaystyle \frac{x_n^0}{q_{nn}}}\left(1-exp\left(-\sqrt{q_{nn}}t\right)\right).\end{array}$$

Proceeding inductively, at the k-th step, one obtains

$$\begin{array}{c}{x}_{n-k}\left(t\right)={(-1)}^k{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^k}}exp\left(-\sqrt{q_{nn}}t\right),\\ x_{n-k-1}\left(t\right)=x_{n-k-1}^0-{(-1)}^{k+1}{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^{k+1}}}\left(1-exp\left(-\sqrt{q_{nn}}t\right)\right),k=\overline{2,n-2},\end{array}$$

where

$$x_{n-k}^0+{(-1)}^{k+1}{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^k}}=0.$$

Hence, the optimal solution of the canonical system is given by

$$x_{n-k}\left(t\right)={(-1)}^k{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^k}}exp\left(-\sqrt{q_{nn}}t\right),k=\overline{0,n-2},$$

(51)

$$x_{n-j}^0+{(-1)}^{j+1}{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^j}}=0,j=\overline{1,n-2},$$

(52)

$$x_1\left(t\right)=x_1^0-{(-1)}^{n-1}{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^{n-1}}}\left(1-exp\left(-\sqrt{q_{nn}}t\right)\right).$$

From (51) it follows that $x_j\left(t\right)\in L_2(0,\infty )$ for $j=\overline{2,n}$, but only for initial data $x_j^0$, $j=\overline{2,n-1}$, satisfying condition (52). The coordinate $x_1\left(t\right)$ converges as $t\to \infty$ to an arbitrary value $c_1\in \mathbb{R}$ according to the formula

$$x_1^0-{(-1)}^{n-1}{\displaystyle \frac{x_n^0}{{\left(\sqrt{q_{nn}}\right)}^{n-1}}}=c_1.$$

(53)

If $c_1\ne 0$, then $x_1\left(t\right)\in C[0,\infty )$; if $c_1=0$, then $x_1\left(t\right)\in L_2(0,\infty )$. In the latter case, the solution described above is referred to as one-coordinate partial stabilization. If conditions (52) are violated, the problem admits no one-coordinate partial solution.

6.1.2. Multi-Coordinate Optimal Stabilization

Suppose the matrix Q has the following structure—$q_{ij}=0,$ $i,j=\overline{1,n-2},$—and the numbers $q_{n-1n-1},$ $q_{nn},$ $q_{n-1n}=q_{nn-1}$ form a positive definite matrix. Further, to reduce the calculations, we assume that $q_{n-1n}=q_{nn-1}=0.$ Then for the coordinates $x_{n-1}\left(t\right),$ $x_n\left(t\right)$ we have a two-dimensional optimal stabilization problem: find the control $u=u[x_{n-1}\left(t\right),x_n\left(t\right)],$ which minimizes the functional

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(q_{n-1n-1}{x}_{n-1}^2\left(t\right)+q_{nn}{x}_n^2\left(t\right)+u^2\left(t\right)\right)dt$$

(54)

subject to the Cauchy problem

$$\begin{array}{c}{\dot{x}}_{n-1}\left(t\right)=x_n\left(t\right),\\ {\dot{x}}_n\left(t\right)=u\left(t\right),\\ x_{n-1}\left(0\right)=x_{n-1}^0,x_n\left(0\right)=x_n^0,\end{array}$$

(55)

where the initial values $x_{n-1}^0$ and $x_n^0$ are fixed but arbitrary.

The system of algebraic Riccati equations takes the form

$$\begin{array}{c}{\left(K_{n-1n}\right)}^2-q_{n-1n-1}=0,\\ K_{n-1n-1}-K_{n-1n}{K}_{nn}=0,\\ 2K_{n-1n}-{\left(K_{nn}\right)}^2+q_{nn}=0.\end{array}$$

(56)

System (56) has a unique positive definite solution. It is worth noting that the optimal stabilization problem (53)–(55) admits a solution also in the case $q_{nn}=0$, i.e., when the weight matrix for the phase coordinates in (53) is merely non-negative definite. The distinction between the positive definite and non-negative definite cases lies solely in the magnitude of the stability margin: it is larger in the former case.

Returning to the case $q_{n-1n-1}>0$, $q_{nn}>0$, the optimal feedback control is

$$u=-K_{nn-1}{x}_{n-1}\left(t\right)-K_{nn}{x}_n\left(t\right).$$

The corresponding solution of the Cauchy problem (55) is

$$\begin{array}{c}{x}_{n-1}\left(t\right)={\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}}}exp\left({\lambda }_{n-1n}^{-}t\right)-{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}}}exp\left({\lambda }_{n-1n}^{+}t\right),\\ x_n\left(t\right)={\lambda }_{n-1n}^{-}{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}}}exp\left({\lambda }_{n-1n}^{-}t\right)-\\ -{\lambda }_{n-1n}^{+}{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}}}exp\left({\lambda }_{n-1n}^{+}t\right),\end{array}$$

(57)

where

$${\lambda }_{n-1n}^{\pm }=-{\displaystyle \frac{\sqrt{q_{nn}+2\sqrt{q_{n-1n-1}}}}{2}}\pm {\displaystyle \frac{\sqrt{q_{nn}-2\sqrt{q_{n-1n-1}}}}{2}}.$$

In Formula (57), the simplifying assumption

$$q_{nn}-2\sqrt{q_{n-1n-1}}>0.$$

The optimal value of the functional (53) is

$$\begin{array}{c}J\left(u\right)=\sqrt{q_{n-1n-1}}\sqrt{q_{nn}+2\sqrt{q_{n-1n-1}}}{\left(x_{n-1}^0\right)}^2+2\sqrt{q_{n-1n-1}}{x}_{n-1}^0{x}_n^0+\\ +\sqrt{q_{nn}+2\sqrt{q_{n-1n-1}}}{\left(x_n^0\right)}^2.\end{array}$$

In this case, the following regularity inclusions hold: $x_{n-1}\left(t\right),x_n\left(t\right)\in L_2(0,\infty ),u\left(t\right)\in L_2(0,\infty ).$ By the canonical structure of system (47), the preceding two coordinates are

$$x_{n-2}\left(t\right)=x_{n-2}^0+{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}\left(exp\left({\lambda }_{n-1n}^{-}t\right)-1\right)-$$

$$-{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\lambda }_{n-1n}^{+}({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+})}}\left(exp\left({\lambda }_{n-1n}^{+}t\right)-1\right),$$

$$x_{n-3}\left(t\right)=x_{n-3}^0+\left(x_{n-2}^0-{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}+{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\lambda }_{n-1n}^{+}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}\right)t+$$

$$+{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\left({\lambda }_{n-1n}^{-}\right)}^2\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}\left(exp\left({\lambda }_{n-1n}^{-}\right)-1\right)-$$

$$-{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\left({\lambda }_{n-1n}^{+}\right)}^2\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}\left(exp\left({\lambda }_{n-1n}^{+}t\right)-1\right).$$

In order to have $x_{n-2}\left(t\right)\in L_2(0,\infty ),$ $x_{n-3}\left(t\right)\in C[0,\infty )$ the initial condition for the coordinate $x_{n-2}\left(t\right)$ must be determined from the equation

$$x_{n-2}^0-{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\lambda }_{n-1n}^{-}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}+{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\lambda }_{n-1n}^{+}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}=0.$$

Thus, the optimal solution of the canonical system (in our case two-coordinate partial stabilization) has the form

$$\begin{array}{c}{x}_{n-k}\left(t\right)={\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\left({\lambda }_{n-1n}^{-}\right)}^{k-1}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}exp\left({\lambda }_{n-1n}^{-}t\right)-\\ -{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\left({\lambda }_{n-1n}^{+}\right)}^{k-1}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}exp\left({\lambda }_{n-1n}^{+}t\right),k=\overline{0,n-1}.\end{array}$$

(58)

In this case, for numbers $2\le j\le n-1$ the following equalities must hold:

$$x_{n-j}^0-{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{+}{x}_{n-1}^0}{{\left({\lambda }_{n-1n}^{-}\right)}^{j-1}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}+{\displaystyle \frac{x_n^0-{\lambda }_{n-1n}^{-}{x}_{n-1}^0}{{\left({\lambda }_{n-1n}^{+}\right)}^{j-1}\left({\lambda }_{n-1n}^{-}-{\lambda }_{n-1n}^{+}\right)}}=0.$$

(59)

From (58) follows the inclusions $x_j\left(t\right)\in L_2(0,\infty ),$ $j=\overline{2,n},$ but not for any $x_j^0,$ $j=\overline{2,n-1},$ only for those that satisfy condition (59). If conditions (59) are violated, the optimal stabilization problem has no two-coordinate partial solution.

From the above considerations it follows that to formulate the problem $(i+1)$-coordinate partial optimal stabilization ($0\le i\le n-2$), it is necessary:

• To formulate the optimal stabilization problem with the objective functional

  $$J_{n-in}\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left({\displaystyle \sum _{l=n-i}^n}{\displaystyle \sum _{k=0}^i}{q}_{n-kl}{x}_{n-k}{x}_l+u^2\left(t\right)\right)dt,$$

  (60)

  subject to the Cauchy problem

  $$\begin{array}{c}{\dot{x}}_{n-i}\left(t\right)=x_{n-i+1}\left(t\right),\\ \dots \dots \dots \dots \dots \dots \\ {\dot{x}}_{n-1}\left(t\right)=x_n\left(t\right),\\ {\dot{x}}_n\left(t\right)=u\left(t\right),\\ x_{n-i}\left(0\right)=x_{n-i}^0.\end{array}$$
  The numbers $q_{kj},k,j=\overline{n-i,n}$ in the criterion (60) form a positive definite symmetric matrix—this is the “strongest” $(i+1)$-coordinate partial stabilization. If in the criterion (60) we limit ourselves only to the positive number $q_{n-in-i},i\ge 2,$ considering all other elements as zero, then we obtain the variant of the “weakest” $(i+1)$-coordinate partial stabilization; we denote the corresponding matrix by $Q^{n-in}$.
• Find the unique positive definite solution of the matrix algebraic Riccati equation

  $${\left(A^{n-in}\right)}^T{K}^{n-in}+K^{n-in}{A}^{n-in}-K^{n-in}{b}^{n-in}{\left(b^{n-in}\right)}^T{K}^{n-in}+Q^{n-in}=0,$$

  where $A^{n-in}$ is the numerical square matrix of dimension $(i+1)\times (i+1),$ in which all elements are 0, and they are 1 above the main diagonal; $b^{n-in}\in R^{i+1}$ and ${\left(b^{n-in}\right)}^T=(0,\dots ,0,1),$ i.e., from number $n-i$ to number $n-1$ there are 0, and in the last position there is 1.
• The optimal stabilizing control has the form

  $$u=-{\displaystyle \sum _{j=n-i}^n}{K}_{nj}^{n-in}{x}_j\left(t\right),$$

  where $x_j\left(t\right),$$j=\overline{n-i,n},$ are solutions of the closed-loop system

  $$\begin{array}{c}{\dot{x}}_{n-i}\left(t\right)=x_{n-i+1}\left(t\right),\\ \dots \dots \dots \dots \dots \dots \\ {\dot{x}}_{n-1}\left(t\right)=x_n\left(t\right),\\ {\dot{x}}_n\left(t\right)=-{\displaystyle \sum _{j=n-i}^n}{K}_{nj}^{n-in}{x}_j\left(t\right),\\ x_j\left(0\right)=x_j^0,j=\overline{n-i,n};\end{array}$$
• Starting from the number $j=n-i+1$ we integrate sequentially the system of differential equations

  $$\begin{array}{c}{\dot{x}}_{n-i+1}\left(t\right)=x_{n-i}\left(t\right),\\ x_{n-i+1}\left(0\right)=x_{n-i+1}^0,i\le n-2;\end{array}$$
• The numbers $x_{n-i+1}^0$ from the previous point are chosen from the condition

  $$x_{n-i+1}\left(t\right)\in L_2(0,\infty ),i\le n-2.$$

6.2. Solving the Critical Optimal Stabilization Problem for a Linear System with One-Dimensional Control

Let us consider the optimal stabilization problem in the critical case, i.e., let the phase vector of the controlled system $x\left(t\right)\in R^n,$$t>0$ satisfy the Cauchy problems

$$\begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),\\ x\left(0\right)=x^0,\end{array}$$

(61)

where $B^T=(b_1,\dots ,b_n).$

The objective functional is (48). We assume that the system (61) is completely controllable; i.e., the columns of the matrix

$$\mathcal{K}=[B,AB,\dots ,A^{n-1}B]$$

are linearly independent. Next, we use the algorithm from [20] to reduce the Cauchy problem (61) to the canonical form in new variables. In our case, the objective functional (48) is also subject to transformation. To this end, choose a vector c orthogonal to $B,AB,\dots ,A^{n-2}B$ and satisfying $(c,A^{n-1}B)=1$, i.e., determine c as the solution of the linear system

$${\mathcal{K}}^{T}c=e_n,$$

(62)

where $e_n^T=(0,\dots ,0,1).$

System (62) has a unique solution since its determinant is nonzero. Introduce the new variable

$$z=Lx,$$

(63)

where the matrix L is defined by

$$L=\left(\begin{array}{c}{c}^T\\ c^{T}A\\ ⋮\\ c^T{A}^{n-1}\end{array}\right).$$

We verify that L is nondegenerate. Suppose

$${\displaystyle \sum _{i=0}^{n-1}}{\alpha }_i{c}^T{A}^i=0.$$

Multiplying this equality successively by $B,AB,\dots ,A^{n-1}B$ and invoking (62), we obtain

$${\alpha }_{n-1}={\alpha }_{n-2}=\dots ={\alpha }_0=0,$$

hence L is nondegenerate.

Let us take the k-th component of the vector z and find its derivative by the system (61)

$${\dot{z}}_k=c^T{A}^{k}x+c^T{A}^{k-1}Bu=c^T{A}^{k}x=z_{k+1},k=\overline{1,n-1},$$

$${\dot{z}}_n=c^T{A}^{n}x+c^T{A}^{n-1}Bu={\displaystyle \sum _{i=1}^n}{p}_i{c}^{\prime }{A}^{i-1}x+u.$$

Here we use the Cayley–Hamilton identity $A^n={\sum }_{i=1}^n{p}_i{A}^{i-1}$, where $p_i$ are the coefficients of the characteristic polynomial of A: ${\lambda }^n-p_n{\lambda }^{n-1}-\dots -p_1=0$.

Consequently, under the substitution (63), system (61) takes the form

$$\begin{array}{c}{\dot{z}}_1=z_2,\\ \dots \dots \dots \dots \dots \\ {\dot{z}}_n=p_1{z}_1+\dots +p_n{z}_n+u=z^T\mathcal{P}+u,\end{array}$$

where $\mathcal{P}={(p_1,\dots ,p_n)}^T.$

Introduce the new control variable

$$v=z^T\mathcal{P}+u.$$

In the new variables $(z,v)$, the objective functional (48) becomes

$$\mathcal{J}\left(v\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(z^T\left(t\right)\mathcal{Q}z\left(t\right)-2z^T\left(t\right)\mathcal{P}v\left(t\right)+v^2\left(t\right)\right)dt,$$

(64)

where

$$\mathcal{Q}={\left(L^{-1}\right)}^{T}Q{L}^{-1}+\mathcal{P}{\left(\mathcal{P}\right)}^T,$$

and the vector $z\left(t\right)$ satisfies the canonical system

$$\begin{array}{c}{\dot{z}}_1=z_2,\\ \dots \dots \dots \dots \dots \\ {\dot{z}}_{n-1}=z_n,\\ {\dot{z}}_n=v\end{array}$$

(65)

with initial conditions

$$z\left(0\right)=z^0=L{x}^0.$$

By the non-degeneracy of $\mathcal{K}$ and L, the criticality property of the original problem is preserved under the coordinate transformation, and we arrive at one of the situations analyzed in the preceding subsection, with the objective functional now containing a quadratic form involving both the state and the control. In the new variables $(z,v)$, the optimal solution of problem (65), (64) is given by the linear feedback

$$v=h^{T}z\left(t\right),h=-\left(KB+\mathcal{P}\right),$$

where the positive definite matrix K is the solution of the Riccati equation

$$A^{T}K+KA+\mathcal{Q}=KB{B}^{T}K+KB{\mathcal{P}}^T+\mathcal{P}{B}^{T}K+\mathcal{P}{\mathcal{P}}^T,$$

$$B^T=(0,\dots ,0,1),A\mathrm{is} \mathrm{the} \mathrm{matrix} \mathrm{of} \mathrm{the} \mathrm{canonical} \mathrm{system}.$$

The optimal value of the objective functional is given by the quadratic form

$$J\left(u\right)={\left(z^0\right)}^{T}K{z}^0.$$

6.3. Solving the Critical Optimal Stabilization Problem for a Linear System with Multidimensional Control

We now consider the optimal stabilization problem in the critical case for a system with multidimensional control; specifically, let the state vector $x\left(t\right)\in {\mathbb{R}}^n$, $t>0$, satisfy the Cauchy problem

$$\begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),\\ x\left(0\right)=x^0,\end{array}$$

(66)

where $B={\left(b_{i,j}\right)}_{i,j=1}^{n,r}.$ The objective functional is

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(x^T\left(t\right)Qx\left(t\right)+u^T\left(t\right)Hu\left(t\right)\right)dt,$$

(67)

where Q is a symmetric non-negative definite matrix of dimension $n\times n$, and H is a symmetric positive definite matrix of dimension $r\times r$.

We assume that system (66) is completely controllable, i.e.,

$$rang\mathcal{K}=rang[B,AB,\dots ,A^{n-1}B]=n.$$

Following the algorithm of [20], we reduce the Cauchy problem, by means of a change of variables and the introduction of new control variables, to r canonical systems each with a scalar control. The objective functional (67) is transformed accordingly.

Without loss of generality, assume that $\mathrm{rank}\left(B\right)=r$. Consider the vectors

$$b_1,b_2,\dots ,b_r,A{b}_1,\dots ,A{b}_r,\dots ,A^{n-1}{b}_1,\dots ,A^{n-1}{b}_r,$$

(68)

where $b_i$ is i-th column of the matrix $B.$

We construct a new sequence of vectors as follows. Start with $b_1,b_2,\dots ,b_r$. Then, examining the vectors of (68) in order, beginning with $A{b}_1$, we check linear independence: if $A{b}_1$ is linearly independent of $b_1,\dots ,b_r$, we append it to the sequence; otherwise, all vectors $A^j{b}_1$, $j\ge 1$, are excluded from further consideration. In general, at each step, the next candidate vector (not previously excluded) is appended to the sequence if it is linearly independent of the current sequence, and otherwise it and all its iterates $A^m\omega$, $m\ge 1$, are discarded. After permuting and possibly renumbering, this procedure yields the sequence

$$b_1,\dots ,A^{n_1-1}{b}_1,\dots ,b_r,\dots ,A^{n_r-1}{b}_r,{\displaystyle \sum _{i=1}^r}{n}_i=n,$$

(69)

where $n_1\ge n_2\ge \dots \ge n_r.$

Choose the vector $c_k$, $k=\overline{1,r}$, such that

$$\left(c_k,A^{n_k-1}{b}_k\right)=1$$

and $c_k$ is orthogonal to all remaining vectors of sequence (69), i.e.,

$$\left(c_k,A^i{b}_j\right)=\left\{\begin{array}{ccc}1,& i=n_k-1,& j=k,\\ 0,& i\ne n_k-1,& j\ne k.\end{array}\right.$$

Let

$$L=\left(\begin{array}{c}{L}_1\\ ⋮\\ L_r\end{array}\right),$$

where

$$L_k=\left(\begin{array}{c}{c}_k^T\\ ⋮\\ c_k^T{A}^{n_k-1}\end{array}\right),k=\overline{1,r}.$$

Perform the coordinate transformation

$$z=Lx.$$

(70)

By the same argument as in the preceding subsection, the matrix L is nondegenerate. Under the substitution (70), system (66) takes the canonical form

$$\begin{array}{c}{\dot{z}}_{s_{i-1}+j}=z_{s_{i-1}+j+1},j=\overline{1,n_i-1},\\ {\dot{z}}_{s_i}={\displaystyle \sum _{j=1}^n}{\tilde{a}}_{s_{i}j}{z}_j+u_i+{\displaystyle \sum _{j=i+1}^r}{m}_{ij}{u}_j,\end{array}$$

(71)

where $s_0=0,$ $s_i={\displaystyle \sum _{k=1}^i}{n}_k,$ $m_{ij}=c_i^T{A}^{n_i-1}{b}_j,$ $j=\overline{i+1,r},$ $i=\overline{1,r}.$

In vector form, system (71) reads

$$\dot{z}=\tilde{A}z+B_{0}Mu,$$

(72)

where $\tilde{A}=LA{L}^{-1}$ is an $n\times n$ matrix whose $s_i$-th row has the form $({\tilde{a}}_{s_{i}1},\dots ,{\tilde{a}}_{s_{i}n})$, $i=\overline{1,r}$; the entries

$${\tilde{a}}_{(s_{i-1}+j)(s_{i-1}+j+1)}=1,j=\overline{1,n_i-1},i=\overline{1,r},$$

and all other elements matrices are zero; $B_0$ is a matrix of dimensions $n\times r,$ in which the i-th element $s_i$-th row is equal to one, $i=\overline{1,r},$ and all other elements of the matrix are zero; M is an upper triangular matrix of dimensions $r\times r$ of the form

$$M=\left(\begin{array}{ccccc}1& c_1^T{A}^{n_1-1}{b}_2& c_1^T{A}^{n_1-1}{b}_3& \dots & c_1^T{A}^{n_1-1}{b}_r\\ 0& 1& c_2^T{A}^{n_2-1}{b}_3& \dots & c_2^T{A}^{n_2-1}{b}_r\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 1\end{array}\right).$$

Introduce the new control variable

$$v=B_0^T\tilde{A}z+Mu.$$

(73)

Under this substitution, system (71) becomes

$$\begin{array}{c}{\dot{z}}_{s_{i-1}+j}=z_{s_{i-1}+j+1},j=\overline{1,n_i-1},\\ {\dot{z}}_{s_i}=v_i,i=\overline{1,r},\end{array}$$

or, in vector form,

$$\dot{z}=A_{0}z+B_{0}v,$$

where $A_0=\mathrm{diag}(A_{01},\dots ,A_{0r})$ is an $n\times n$ block-diagonal matrix and $A_{0i}$ is the $n_i\times n_i$ nilpotent Jordan block (ones on the first superdiagonal, zeros elsewhere).

From (70) and (73) we obtain

$$x=L^{-1}z,u=M^{-1}\left(v-B_0^T\tilde{A}z\right).$$

The objective functional (67) then transforms to

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(z^T\left(t\right)\mathcal{Q}z\left(t\right)-2z^T\left(t\right)\mathcal{G}v\left(t\right)+v^T\left(t\right)\mathcal{H}v\left(t\right)\right)dt,$$

where

$$\mathcal{Q}={\left(L^{-1}\right)}^{T}Q{L}^{-1}+{\tilde{A}}^T{B}_0{\left(M^{-1}\right)}^{T}H{M}^{-1}{B}_0^T\tilde{A},$$

$$\mathcal{G}={\tilde{A}}^T{B}_0{\left(M^{-1}\right)}^{T}H{M}^{-1},\mathcal{H}={\left(M^{-1}\right)}^{T}H{M}^{-1}.$$

By the non-degeneracy of the transformations introduced above, the original critical optimal stabilization problem decomposes into r independent one-dimensional subproblems for the systems

$$\begin{array}{c}{\dot{z}}_{s_{i-1}+j}=z_{s_{i-1}+j+1},j=\overline{1,n_i-1},\\ {\dot{z}}_{s_i}=v_i,\end{array}$$

with objective functional

$$\begin{array}{c}{J}_{s_i}\left(u_i\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(z_{s_i}^T\left(t\right){\mathcal{Q}}_{s_i}{z}_{s_i}\left(t\right)-2z_{s_i}^T\left(t\right){\mathcal{G}}_{s_i}{v}_{s_i}\left(t\right)+v_{s_i}^T\left(t\right){\mathcal{H}}_{s_i}{v}_{s_i}\left(t\right)\right)dt,i=\overline{1,r},\\ J\left(u\right)={\displaystyle \sum _{i=1}^r}{J}_{s_i}\left(u_i\right),\end{array}$$

(74)

where all vectors and matrices in (74) are determined by the transformations described above.

To illustrate the details of the algorithm, we consider an example. Although the matrices in this example are chosen to demonstrate the mathematical efficiency of the proposed decomposition algorithm, this 3D state-space structure has a clear physical analogy. Specifically, in the context of unmanned aerial vehicles (UAVs), these variables can model the pitch, roll, and yaw angles, where the critical imaginary eigenvalues represent undamped aerodynamic oscillations. The developed algorithm successfully stabilizes these critical states where standard LQR controllers fail.

Consider the controlled system from [20]:

$$\begin{array}{c}{\dot{x}}_1=x_1+x_2+x_3+u_1+u_2,\\ {\dot{x}}_2=x_1-x_2+x_3,\\ {\dot{x}}_3=-x_1+x_2-x_3-u_2,\end{array}$$

(75)

with initial conditions

$$x_i\left(0\right)=x_i^0,i=\overline{1,3}.$$

where the matrices A and B are

$$A=\left(\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ -1& 1& -1\end{array}\right),B=\left(\begin{array}{cc}1& 1\\ 0& 0\\ 0& -1\end{array}\right).$$

The sequence of vectors (69) takes the form

$$b_1=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),A{b}_1=\left(\begin{array}{c}1\\ 1\\ -1\end{array}\right),b_2=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right),n_1=2,n_2=1,rang\mathcal{K}=3.$$

On the solutions of system (75), we consider the objective functional

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(x_2^2\left(t\right)+x_3^2\left(t\right)+u_1^2\left(t\right)+u_2^2\left(t\right)\right)dt.$$

(76)

Let us choose the vectors $c_1^T=\left(c_1^1,c_2^1,c_3^1\right)$ and $c_2^T=\left(c_1^2,c_2^2,c_3^2\right)$ from the conditions

$$\left\{\begin{array}{c}\left(c_1,b_1\right)=0,\hfill \\ \left(c_1,A{b}_1\right)=1,\hfill \\ \left(c_1,b_2\right)=0,\hfill \end{array}\right.\left\{\begin{array}{c}\left(c_2,b_1\right)=0,\hfill \\ \left(c_2,A{b}_1\right)=0,\hfill \\ \left(c_2,b_2\right)=1.\hfill \end{array}\right.$$

The written systems have solutions $c_1^T=(0,1,0),$ $c_2^T=(0,-1,-1).$ Then the change of variables has the form

$$z=Lx=\left(\begin{array}{c}{c}_1^T\\ c_1^{T}A\\ c_2^T\end{array}\right)x=\left(\begin{array}{ccc}0& 1& 0\\ 1& -1& 1\\ 0& -1& -1\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right).$$

Furthermore,

$$L^{-1}=\left(\begin{array}{ccc}2& 1& 1\\ 1& 0& 0\\ -1& 0& -1\end{array}\right).$$

Then the system (75) in the new variables takes the form (72), i.e.,

$$\dot{z}=\tilde{A}z+B_{0}Mu,$$

where

$$\tilde{A}=LA{L}^{-1}=\left(\begin{array}{ccc}0& 1& 0\\ 2& -1& 0\\ 0& 0& 0\end{array}\right),$$

$$B_0=\left(\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right),M=\left(\begin{array}{cc}1& c_1^{T}A{b}_2\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).$$

Thus, in the new variables we obtain the system

$$\left\{\begin{array}{c}{\dot{z}}_1=z_2,\hfill \\ {\dot{z}}_2=2z_1-z_2+u_1,\hfill \\ {\dot{z}}_3=u_2\hfill \end{array}\right.$$

Introducing the substitution

$$v=\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)=\left(\begin{array}{c}2z_1-z_2+u_1,\\ u_2\end{array}\right),$$

system (75) decomposes into two independent controlled subsystems:

$$\begin{array}{c}\left\{\begin{array}{c}{\dot{z}}_1=z_2,\hfill \\ {\dot{z}}_2=v_1,\hfill \end{array}\right.\\ z_1\left(0\right)=z_1^0,z_2\left(0\right)=z_2^0,\end{array}$$

$${\dot{z}}_3=v_2,z_3\left(0\right)=z_3^0.$$

The functional (76) becomes

$$J\left(u\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(x_2^2\left(t\right)+x_3^2\left(t\right)+u_1^2\left(t\right)+u_2^2\left(t\right)\right)dt=$$

$$={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(z_2^2\left(t\right)+{(z_1\left(t\right)+z_3\left(t\right))}^2+{(v_1\left(t\right)-2z_1\left(t\right)+z_2\left(t\right))}^2+v_2^2\left(t\right)\right)dt=J_{s_1}\left(v_1\right)+J_{s_2}\left(v_2\right).$$

Thus, the original optimal stabilization problem decomposes into two independent subproblems:

(1) Minimize the functional

$$J_{s_2}\left(v_2\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(z_3^2\left(t\right)+v_2^2\left(t\right)\right)dt,$$

subject to the system

$${\dot{z}}_3=v_2,z_3\left(0\right)=z_3^0;$$

(2) Minimize the functional

$$J_{s_1}\left(v_1\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(z_2^2\left(t\right)+z_1^2\left(t\right)+2z_1\left(t\right)z_3\left(t\right)+{(v_1\left(t\right)-2z_1\left(t\right)+z_2\left(t\right))}^2\right)dt$$

subject to the system

$$\left\{\begin{array}{c}{\dot{z}}_1=z_2,\hfill \\ {\dot{z}}_2=v_1,\hfill \end{array}\right.$$

$$z_1\left(0\right)=z_1^0,z_2\left(0\right)=z_2^0.$$

7. Discussion and Conclusions

This paper addresses the critical case of linear–quadratic optimal stabilization, in which the system matrix possesses purely imaginary eigenvalues and the weight matrices in the objective functional are not required to be strictly positive definite. Necessary and sufficient conditions for the existence of a solution are established: the weight matrix of the functional must penalize precisely those state components that correspond to the imaginary eigenvalues of the system matrix.

Two regularization approaches are presented and analyzed: perturbation of the functional weight matrix $Q^{\epsilon }=Q+{\epsilon }^2{E}_n$ and perturbation of the system matrix $A^{\epsilon }=A+\epsilon E_n$. Both methods yield a stabilising control that, as $\epsilon \to 0$, converges to the solution of the critical problem with accuracy of order $O\left(\epsilon \right)$. The resulting error estimates provide a practical tool for tuning controllers in applications.

A constructive algorithm for decomposing multidimensional critical problems into a family of one-dimensional canonical subproblems is also developed.

The results obtained constitute a theoretical foundation for the design of robust control systems in robotics, aerospace engineering, and related fields, where physical parameters may vary under external influences, driving the system into critical operating regimes. From a theoretical standpoint, the developed regularization framework successfully extends the classical linear–quadratic control theory to non-standard, degenerate cases where traditional algebraic Riccati equations lack standard solutions.

Future work may focus on extending the present results to stochastic systems and systems with impulsive perturbations. The algorithms for reduction to canonical form developed here may also serve as a computational tool for the design of complex controlled systems within specialized application software.