Necessary Conditions for the Optimality and Sustainability of Solutions in Infinite-Horizon Optimal Control Problems

Abstract

The paper deals with an infinite-horizon optimal control problem with general asymptotic endpoint constraints. The fulfillment of constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions. A new version of the Pontryagin maximum principle with an explicitly specified adjoint variable is developed. The proof of the main results is based on the fact that the restriction of the optimal process to any finite time interval is a solution to the corresponding finite-horizon problem containing the conditional cost of the phase vector as a terminal term.

1. Introduction

Infinite-horizon optimal control problems arise in numerous applications such as economics, engineering, ecology, and biology [1]. However, the theory of the necessary optimality conditions for these problems is less developed than in the finite-horizon case [2]. In the present paper, we develop new necessary optimality conditions for infinite-horizon optimal control problems with initial and general asymptotic endpoint constraints. Our interest in the problem is motivated mainly by economic applications. When studying different optimal economic growth models [3,4], the question about the sustainability of optimal growth [5,6] arises naturally. In the general case, optimality does not guarantee sustainability. The optimality of growth is usually understood in economics in the sense of the maximization of a utility functional that characterizes per capita consumption over an infinite time horizon. Meanwhile, the sustainability of growth means the fulfillment of certain asymptotic conditions that characterize per capita consumption and/or environmental quality in the long-run perspective. Combining the concepts of the optimality and sustainability of growth naturally leads to optimal control problems with asymptotic endpoint constraints [7].

There is a lot of literature on different aspects of the infinite-horizon optimal control theory and its applications in economics (see, for example, [1,8,9,10,11,12] and the references therein). However, to the best of the author’s knowledge, only a few papers are concerned with the development of the necessary optimality conditions for problems with asymptotic endpoint constraints [13,14,15]. Moreover, the developed necessary conditions use a rather restrictive a priori assumption that admissible trajectories converge at infinity. This circumstance can be explained by technical obstacles arising in considerations of optimal control problems on infinite time intervals. Infinite time horizon introduces a singularity in the problems that produce different pathologies in the relations of the maximum principle [8,9,16]. The situation becomes even more challenging in the case of problems with asymptotic endpoint constraints without a priori assumptions about the behaviors of admissible trajectories.

In the present paper, under certain controllability-type assumptions, the problem with general asymptotic endpoint constraints under consideration is reduced to a family of standard finite-horizon problems with utility functionals containing the value of the conditional cost of the phase vector as a terminal term. Using this approach, a new version of the Pontryagin maximum principle containing an explicit characterization of the adjoint variable is developed, without any a priori assumptions about the asymptotic behavior of admissible trajectories. The developed approach is based on the use of the conditional cost function. The conditional cost function has an economic sense of aggregated intertemporal utility gained when the system moves from a given initial state on an infinite time interval, provided that the control is fixed [17]. The results obtained generalize and enhance the previous results of the author in this direction [18,19]. The main novelty is that general asymptotic endpoint constraints are tackled. Asymptotic constraints of this type can be viewed as the minimal necessary conditions for the sustainability of solutions ([7], Part II). Sufficient conditions for the continuous differentiability of the conditional cost function are also presented. A simple model of the optimal sustainable exploitation of a non-renewable resource is considered as an illustrative example.

2. Statement of the Problem

Let a non-empty open convex set G from ${\mathbb{R}}^n$, non-empty closed sets $M_0\subset G$ and $M_{\infty }\subset {\mathbb{R}}^n$, functions

$$f:[0,\infty )\times G\times {\mathbb{R}}^m\to {\mathbb{R}}^n,f^0:[0,\infty )\times G\times {\mathbb{R}}^m\to {\mathbb{R}}^1,\phi :G\to {\mathbb{R}}^1$$

and a multivalued mapping $U:[0,\infty )\rightrightarrows {\mathbb{R}}^m$ with non-empty values be given.

In the following, we assume that for a.e. $t\in [0,\infty )$, the derivatives $f_x(t,x,u)$ and $f_x^0(t,x,u)$ exist for all $(x,u)\in G\times {\mathbb{R}}^m$, and the functions $f(·,·,·)$, $f^0(·,·,·)$, $f_x(·,·,·)$, and $f_x^0(·,·,·)$ are Lebesgue–Borel-measurable ($LB$-measurable) in $(t,u)$ for any $x\in G$ ([20], Definition 6.33) and continuous in x for a.e. $t\in [0,\infty )$ and all $u\in {\mathbb{R}}^m$. We also assume that the multivalued mapping $U(·)$ is $LB$-measurable. The latter means that its graph, i.e., the set $\mathrm{graph}U(·)=\{(t,u)\in [0,\infty )\times {\mathbb{R}}^m:u\in U\left(t\right)\}$, is an $LB$-measurable set in ${\mathbb{R}}^{m+1}$. Concerning the function $\phi (·)$, we assume that it is locally Lipschitz continuous.

Consider the following problem $\left(P\right)$:

$$J(x(·),u(·))=\phi \left(x\left(0\right)\right)+{\int }_0^{\infty }{f}^0(t,x\left(t\right),u\left(t\right))dt\to \max ,$$

(1)

$$\dot{x}\left(t\right)=f(t,x\left(t\right),u\left(t\right)),$$

(2)

$$u\left(t\right)\in U\left(t\right),$$

(3)

$$x\left(0\right)\in M_0,\underset{t\to \infty }{lim\; inf}\rho (x\left(t\right),M_{\infty })=0.$$

(4)

Here, $x\left(t\right)=(x^1\left(t\right),\cdots ,x^n\left(t\right))\in {\mathbb{R}}^n$ and $u\left(t\right)=(u^1\left(t\right),\cdots ,u^m\left(t\right))\in {\mathbb{R}}^m$ are values of the state vector and the control at the instant of time $t\ge 0$, respectively, and $\rho (a,M)=\min {}_{\xi \in M}\parallel a-\xi \parallel$ is the distance from the point $a\in {\mathbb{R}}^n$ to a non-empty closed set $M\subset {\mathbb{R}}^n$.

Note that the asymptotic endpoint constraint in (4) does not assume any a priori conditions on the asymptotic behavior of the state variable $x(·)$. In economic applications, the validity of asymptotic constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions ([7], Part II).

An arbitrary Lebesgue measurable function $u:[0,\infty )\to {\mathbb{R}}^m$, satisfying the inclusion (3) for all $t\ge 0$, is called a control. For a given control $u(·)$, the corresponding trajectory$x(·)$ is a locally absolutely continuous solution of the differential Equation (2), which is defined in G on some finite or infinite time interval $[0,\tau )$, $\tau >0$ (if such a solution exists). The local absolute continuity of the solution $x(·)$ means that the function $x(·)$ is absolutely continuous on any finite interval $[0,T]$, $T>0$, in its domain $[0,\tau )$.

The pair $\left(x\right(·),u(·\left)\right)$, where $u(·)$ is a control and $x(·)$ is the corresponding trajectory, is called a process if the trajectory $x(·)$ is defined in G on the infinite interval $[0,\infty )$. The pair $\left(x\right(·),u(·\left)\right)$ is called an admissible process or an admissible pair in $\left(P\right)$ if the trajectory $x(·)$ is defined on the interval $[0,\infty )$ in G and satisfies the endpoint constraints (4) and the function $t\mapsto f^0(t,x\left(t\right),u\left(t\right))$ is locally integrable on $[0,\infty )$. Thus, for any admissible process $\left(x\right(·),u(·\left)\right)$ and any $T>0$, the integral ${\int }_0^T{f}^0(t,x\left(t\right),u\left(t\right))dt$ is defined.

Since the functional (1) involves an improper integral, different concepts of the optimality of an admissible process $(x_{*}(·),u_{*}(·))$ in the problem $\left(P\right)$ can be used. In the following, we employ the concept of weak overtaking optimality [1].

Definition 1. 

An admissible process $(x_{*}(·),u_{*}(·))$ is called weakly overtaking optimal in the problem $\left(P\right)$ if for any other admissible process $\left(x\right(·),u(·\left)\right)$, the following inequality holds:

$$\phi \left(x_{*}\left(0\right)\right)-\phi \left(x\left(0\right)\right)+\underset{T\to \infty }{lim\; sup}\left[{\int }_0^T{f}^0(t,x_{*}\left(t\right),u_{*}\left(t\right))dt-{\int }_0^T{f}^0(t,x\left(t\right),u\left(t\right))dt\right]\ge 0.$$

In general, the concept of weak overtaking optimality is applicable in the case when the improper integral in (1) does not necessarily converge on the pair $(x_{*}(·),u_{*}(·))$. However, even in the case when the improper integral in (1) converges on $(x_{*}(·),u_{*}(·))$, this concept is weaker than the standard concept of strong optimality [1].

For a process $(x_{*}(·),u_{*}(·))$ (not necessarily admissible), we use the following weak regularity condition [9,21].

(A1)  There exists a continuous function $\gamma :[0,\infty )\to (0,\infty )$ and a locally integrable function $\phi :[0,\infty )\to {\mathbb{R}}^1$, such that $\{x:\parallel x-x_{*}\left(t\right)\parallel \le \gamma \left(t\right)\}\subset G$ for all $t\ge 0$ and a.e. $t\in [0,\infty )$

$$\max {}_{\{x:\parallel x-x_{*}\left(t\right)\parallel \le \gamma \left(t\right)\}}\left\{\parallel f_x(t,x,u_{*}\left(t\right))\parallel +\parallel f_x^0(t,x,u_{*}\left(t\right))\parallel \right\}\le \phi \left(t\right).$$

To the best of the author’s knowledge, the first-order necessary optimality conditions in the form of the Pontryagin maximum principle for problems on finite time intervals with LB-measurable data under weak regularity conditions were developed by F. Clarke in the 1970s (see [20,22] for more details).

As shown in [21], the fulfillment of condition $\left(A1\right)$ for the process $(x_{*}(·),u_{*}(·))$ ensures the applicability of standard theorems on finite time intervals on the existence and continuous dependence of a solution $x(\tau ,\xi ;·)$ of the Cauchy problem

$$\dot{x}\left(t\right)=f(t,x\left(t\right),u_{*}\left(t\right)),x\left(\tau \right)=\xi .$$

(5)

on the initial data $(\tau ,\xi )$ ([23], $\S 2.5.5$, Theorem 1), and the theorem on the differentiability of the solution $x(\tau ,\xi ;·)$ of the Cauchy problem (5) with respect to the initial value $\xi$ for all $\tau \ge 0$ ([23], $\S 2.5.6$). In the following, for brevity, in the case $\tau =0$, the first argument in the notation $x(0,\xi ;·)$ is sometimes omitted, and instead of $x(0,\xi ;·)$, we simply write $x(\xi ;·)$.

For a process $(x_{*}(·),u_{*}(·))$ that satisfies condition $\left(A1\right)$, we denoted its normalized fundamental matrix solutions at time zero as $Y_{*}(·)$ and $Z_{*}(·)$ for the linear systems

$$\dot{y}\left(t\right)=f_x(t,x_{*}\left(t\right),u_{*}\left(t\right))y\left(t\right)$$

and

$$\dot{z}\left(t\right)=-{\left[f_x(t,x_{*}\left(t\right),u_{*}\left(t\right))\right]}^{*}z\left(t\right)$$

(6)

respectively. By virtue of $\left(A1\right)$, both matrix functions $Y_{*}(·)$ and $Z_{*}(·)$ are defined on the entire interval $[0,\infty )$, and in addition, ${\left[Z_{*}\left(t\right)\right]}^{-1}\equiv {\left[Y_{*}\left(t\right)\right]}^{*}$, $t\ge 0$.

Thus, if $(x_{*}(·),u_{*}(·))$ is a process that satisfies condition $\left(A1\right)$, then the function $t\mapsto f_x(t,x_{*}\left(t\right),u_{*}\left(t\right))$ is locally integrable on the interval $[0,\infty )$. For arbitrary times $0\le \tau <s$, according to the theorem on the continuous dependence of the solution of the Cauchy problem on the initial data, for each vector $\xi$ from some neighborhood $\mathcal{V}\left(x_{*}\left(\tau \right)\right)\subset G$ of the point $x_{*}\left(\tau \right)$, a unique solution $x(\tau ,\xi ;·)$ of the Cauchy problem (5) is defined on the interval $[0,s]$. According to the theorem on the differentiability of the solution of the Cauchy problem with respect to the initial value, for any $t\ge 0$, the function $x(\tau ,·;t):\mathcal{V}\left(x_{*}\left(\tau \right)\right)\to {\mathbb{R}}^n$ is continuously differentiable and

$$x_{\xi }(\tau ,x_{*}\left(\tau \right);s))=Y_{*}\left(s\right)Y_{*}^{-1}\left(\tau \right)={\left[Z_{*}\left(\tau \right){\left[Z_{*}\left(s\right)\right]}^{-1}\right]}^{*}.$$

(7)

The following growth condition for a process $(x_{*}(·),u_{*}(·))$ was introduced in [24] as a generalization of the dominating discount condition (see [9] for details).

(A2)  There exists a number $\beta >0$ and an integrable function $\lambda :[0,\infty )\mapsto {\mathbb{R}}^1$, such that for any $\zeta \in G$, $\parallel \zeta -x_{*}\left(0\right)\parallel <\beta$, the solution $x(\zeta ;·)$ of the Cauchy problem for the differential Equation (2) with $u(·)=u_{*}(·)$ and the initial condition $x\left(0\right)=\zeta$ exists in G on the interval $[0,\infty )$ and

$$\max {}_{\theta \in [x(\zeta ;t),x_{*}\left(t\right)]}\left|\langle f_x^0(t,\theta ,u_{*}\left(t\right)),x(\zeta ;t)-x_{*}\left(t\right)\rangle \right|\stackrel{\mathrm{a}.\mathrm{e}.}{\le }\parallel \zeta -x_{*}\left(0\right)\parallel \lambda \left(t\right).$$

Here, $[x(\zeta ;t),x_{*}\left(t\right)]$ is a segment with vertices $x(\zeta ;t)$ and $x_{*}\left(t\right)$ in ${\mathbb{R}}^n$.

The proof of the following results can be found in ([21], Lemma 3.2).

Lemma 1. 

Let the process $(x_{*}(·),u_{*}(·))$ satisfy the conditions $\left(A1\right)$ and $\left(A2\right)$. Then, the following estimate is valid:

$$\parallel {\left[Z_{*}\left(t\right)\right]}^{-1}{f}_x^0(t,x_{*}\left(t\right),u_{*}\left(t\right))\parallel \le \sqrt{n}\lambda \left(t\right)\mathrm{a}.\mathrm{e}.t\ge 0.$$

If both conditions $\left(A1\right)$ and $\left(A2\right)$ are satisfied, then according to Lemma 1, one can define the function $\psi :[0,\infty )\to {\mathbb{R}}^n$ using the equality

$$\psi \left(t\right)=Z_{*}\left(t\right){\int }_t^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,t\ge 0.$$

(8)

As shown in [21] (see also [9]), in the case of the infinite-horizon problem $\left(P\right)$ without asymptotic endpoint constraints (i.e., $M_{\infty }={\mathbb{R}}^n$) and with a fixed initial state $x\left(0\right)=x_0$, if the conditions $\left(A1\right)$ and $\left(A2\right)$ hold, then the weakly overtaking admissible process $(x_{*}(·),u_{*}(·))$ satisfies the normal form maximum principle with an adjoint function $\psi (·)$ specified in formula (8).

Below we employ the following growth condition, which is obviously a stronger variant of $\left(A2\right)$.

(${\mathrm{A}2}^{\prime }$)  There is a number $\beta >0$ and an integrable function $\lambda :[0,\infty )\to {\mathbb{R}}^1$, such that for any $\zeta \in G$, $\parallel \zeta -x_{*}\left(0\right)\parallel <\beta$, the solution $x(\zeta ;·)$ of the Cauchy problem for the differential Equation (2) with $u(·)=u_{*}(·)$ and the initial condition $x\left(0\right)=\zeta$ exists in G on the interval $[0,\infty )$. Moreover, for any ${\zeta }_i$, $\parallel {\zeta }_i-x_{*}\left(0\right)|<\beta$, $i=1,2$, the following inequality holds:

$$\max {}_{\theta \in [x({\zeta }_1;t),x({\zeta }_2;t)]}\left|\langle f_x^0(t,\theta ,u_{*}\left(t\right)),x({\zeta }_1;t)-x({\zeta }_2;t)\rangle \right|\stackrel{a.e.}{\le }\parallel {\zeta }_1-{\zeta }_2\parallel \lambda \left(t\right).$$

Here, $[x({\zeta }_1;t),x({\zeta }_2;t)]$ is a segment with vertices $x({\zeta }_1;t)$ and $x({\zeta }_2;t)$ in ${\mathbb{R}}^n$.

For the given controls $u(·)$ and $u_{*}(·)$, and $\tau >0$, the corresponding composite control $u_{*}^{\tau }(·)$ is defined as follows:

$$u_{*}^{\tau }\left(t\right)=\left\{\begin{array}{cc}u\left(t\right),\hfill & t\in [0,\tau ),\hfill \\ u_{*}\left(t\right),\hfill & t\in [\tau ,\infty ).\hfill \end{array}\right.$$

(9)

The following condition characterizes the stability of the sustainability property of an admissible process $(x_{*}(·),u_{*}(·))$ with respect to small perturbations of the trajectory $x_{*}(·)$ in a neighborhood of the set $M_{\infty }$.

(A3)  There exists $\epsilon >0$, such that if an admissible process $(x_{*}(·),u_{*}(·))$ satisfies the inequality $\rho (x_{*}\left(\tau \right),M_{\infty })\le \epsilon$ at an instant $\tau \ge 0$, then there is $\delta (\tau ,x_{*}\left(\tau \right))>0$, such that for any solution $x(·)$ to Equation (2) corresponding to a control $u(·)$ on the interval $[0,\tau ]$ in G with an initial state $x\left(0\right)\in M_0$ and satisfying the conditions $\parallel x\left(\tau \right)-x_{*}\left(\tau \right)\parallel \le \delta (\tau ,x_{*}\left(\tau \right))$ and $\rho (x\left(\tau \right),M_{\infty })\le \rho (x_{*}\left(\tau \right),M_{\infty })$, the pair $(x_{*}^{\tau }(·),u_{*}^{\tau }(·))$ is an admissible process in $\left(P\right)$. Here, $u_{*}^{\tau }(·)$ is the composite control corresponding to $u(·)$, $u_{*}(·)$, and τ (see (9)), and $x_{*}^{\tau }(·)$ is the corresponding trajectory.

Note that if there are no asymptotic endpoint constraints in $\left(P\right)$ (i.e., $M_{\infty }={\mathbb{R}}^n$) and $\left(A2\right)$ is satisfied for an admissible process $(x_{*}(·),u_{*}(·))$, then the condition $\left(A3\right)$ automatically holds for the process $(x_{*}(·),u_{*}(·))$.

3. Conditional Cost Function

Let $(x_{*}(·),u_{*}(·))$ be a process for which the condition $\left(A{2}^{\prime }\right)$ is satisfied. Then, any solution $x(\zeta ;·)$ of the Cauchy problem (5) with the initial state $\zeta$ lying in the $\beta$-neighborhood of the point $x_0$, i.e., $\parallel \zeta -x_0\parallel <\beta$, is defined in G on the entire interval $[0,\infty )$. Therefore, the pair $(x(\zeta ;·),u_{*}(·))$ is also a process. In the following, we assume that the process $(x_{*}(·),u_{*}(·))$ is fixed, and for any process $(x(\zeta ;·),u_{*}(·))$ with the initial state $\zeta$ from the $\beta$-neighborhood of the point $x_0$, the condition $\left(A1\right)$ is satisfied (for each process $(x(\zeta ;·),u_{*}(·))$ with its own functions $\gamma (·)$ and $\varphi (·)$). In this case, according to condition $\left(A1\right)$, the trajectory $x(\zeta ;·)$ is uniquely defined ([25], Chapter 1, Theorem 2).

Define the set $\Omega \subset [0,\infty )\times {\mathbb{R}}^n$ using the following equality:

$$\Omega =\bigcup _{\zeta :\parallel \zeta -x_0\parallel <\beta }\{(\tau ,\xi )\in {\mathbb{R}}^{n+1}:\xi =x(\zeta ;\tau ):\tau \ge 0\}.$$

(10)

Lemma 2. 

The set Ω defined by the equality (10) is open in $[0,\infty )\times G$.

Proof. 

Let $(\tilde{\tau },\tilde{\xi })\in \Omega$. Then, for some $\tilde{\zeta }$ from the $\beta$-neighborhood of the point $x_0$, we have $\tilde{\xi }=x(\tilde{\zeta };\tilde{\tau })$ (see (10)). Let us choose a sufficiently small $\tilde{\beta }>0$, such that the $\tilde{\beta }$-neighborhood of the point $\tilde{\zeta }$ is contained in the $\beta$-neighborhood of the point $x_0$. According to the condition $\left(A1\right)$ and the theorem on the existence and continuous dependence of the solution of the Cauchy problem from the initial data, there is $\tilde{\epsilon }>0$, such that for any point $(\tau ,\xi )$ from the $\tilde{\epsilon }$-neighborhood $\Omega (\tilde{\tau },\tilde{\xi })$ of the point $(\tilde{\tau },\tilde{\xi })$ in $[0,\infty )\times G$, the trajectory $x(\tau ,\xi ;·)$ is defined on the interval $[0,\tau ]$ and satisfies the inequality $\parallel x(\tau ,\xi ;0)-\tilde{\zeta }\parallel <\tilde{\beta }$ and hence $\parallel x(\tau ,\xi ;0)-x_0\parallel <\beta$. Therefore, $\Omega (\tilde{\tau },\tilde{\xi })\subset \Omega$ (see (10)). Hence, the set $\Omega$ is open in $[0,\infty )\times G$. □

Let $(\tau ,\xi )\in \Omega$ and $0\le \tau <s$. Define the intertemporal instantaneous utility function $\pi (\tau ,·,s)$ on the open set

$$G_{\tau }=\{x\in G:(\tau ,x)\in \Omega \}$$

(11)

as follows:

$$\pi (\tau ,\xi ,s)=f^0(s,x(\tau ,\xi ;s),u_{*}\left(s\right)),\xi \in G_{\tau }.$$

(12)

Substantially, the value $\pi (\tau ,\xi ,s)$ is the instantaneous utility gained at the moment s, depending on the vector $\xi \in G_{\tau }$ at the previous moment $\tau$, provided that the system moves along the trajectory $x(\tau ,\xi ;·)$ with the fixed control $u_{*}(·)$ on the interval $[\tau ,s]$.

Suppose now that the improper integral in (1) converges on the process $(x_{*}(·),u_{*}(·))$. Let us show that according to the condition $\left(A{2}^{\prime }\right)$, the integral

$$W(\tau ,\xi )={\int }_{\tau }^{\infty }\pi (\tau ,\xi ,s)ds$$

(13)

also converges for any $(\tau ,\xi )\in \Omega$ in this case. Indeed, for any $T>0$ we have

$$\begin{array}{c}{\int }_{\tau }^T\pi (\tau ,\xi ,s)ds={\int }_{\tau }^T{f}^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds\hfill \\ \hfill +{\int }_{\tau }^T\left[f^0(s,x(\tau ,\xi ;s),u_{*}\left(s\right))-f^0(s,x_{*}\left(s\right),u_{*}\left(s\right))\right]ds.\end{array}$$

According to $\left(A{2}^{\prime }\right)$, for a.e. $s\ge 0$ we have

$$\left|f^0(s,x(\tau ,\xi ;s),u_{*}\left(s\right))-f^0(s,x_{*}\left(s\right),u_{*}\left(s\right))\right|\le \parallel x(\tau ,\xi ;0)-x_{*}\left(0\right)\parallel \lambda \left(s\right)$$

where $\lambda (·)$ is an integrable function on $[0,\infty )$. This implies that the limit

$$W(\tau ,\xi )={\int }_{\tau }^{\infty }\pi (\tau ,\xi ,s)ds=\underset{T\to \infty }{lim}{\int }_t^T\pi (\tau ,\xi ,s)ds$$

exists and is finite. Thus, for any $(\tau ,\xi )\in \Omega$, the value $W(\tau ,\xi )$ is well-defined (see (12) and (13)).

Substantially, the value $W(\tau ,\xi )$ is the conditional cost of the phase vector $\xi \in G$ at the moment $\tau \ge 0$, i.e., it is the aggregate intertemporal utility gained when the system moves from the state $\xi$ at instant $\tau$ on the interval $[\tau ,\infty )$, provided that the control $u_{*}(·)$ is used. The function $W(·,·)$ defined on the set $\Omega$ using the equality (13) is called the conditional cost function. Note that the conditional cost function was introduced in the author’s paper [17] in the context of the economic interpretation of the adjoint variable in relation to the Pontryagin maximum principle for infinite-horizon problems. In this paper, this function is used to develop a new version of the Pontryagin maximum principle for the problem $\left(P\right)$. The main advantage of the usage of the conditional cost function $W(·,·)$ compared to the optimal value function $V(·,·)$, $V(\tau ,\xi )={sup}_{\left(x\right(·),u(·\left)\right),x\left(\tau \right)=\xi }J(x(·),u(·))$, $\tau \ge 0$, $\xi \in G$, is its differentiability under rather mild assumptions (see Lemma 3 below).

For $(\tau ,\xi )\in \Omega$ and the process $(x(\tau ,\xi ;·),u_{*}(·))$, we denote (analogously to $Y_{*}(·)$ and $Z_{*}(·)$) as $Y(\tau ,\xi ;·)$ and $Z(\tau ,\xi ;·)$ the normalized fundamental matrix solutions at time zero of the linear systems

$$\dot{y}\left(t\right)=f_x(t,x(\tau ,\xi ;t),u_{*}\left(t\right))y\left(t\right)$$

and

$$\dot{z}\left(t\right)=-{\left[f_x(t,x(\tau ,\xi ;t),u_{*}\left(t\right))\right]}^{*}z\left(t\right)$$

respectively. Since $(x(\tau ,\xi ;·),u_{*}(·))$ is a process, the functions $Y(\tau ,\xi ;·)$ and $Z(\tau ,\xi ;·)$ are defined on the entire time interval $[0,\infty )$, and ${\left[Z(\tau ,\xi ,t)\right]}^{-1}\equiv {\left[Y(\tau ,\xi ;t)\right]}^{*}$, $t\ge 0$.

Lemma 3. 

Let $(x_{*}(·),u_{*}(·))$ be a process for which the condition $\left(A{2}^{\prime }\right)$ holds, and assume that the value $J(x_{*}(·),u_{*}(·))$ is finite. Assume also that for any process $(x(\zeta ;·),u_{*}(·))$ with the initial state ζ from the β-neighborhood of the point $x_0$, the condition $\left(A1\right)$ is satisfied. Then, for any $\tau \ge 0$, the function $\xi \mapsto W(\tau ,\xi )$ is continuously differentiable on the set $G_{\tau }$ (see (11)) and

$$W_{\xi }(\tau ,\xi )={\int }_{\tau }^{\infty }{\pi }_{\xi }(\tau ,\xi ,s)ds=Z(\tau ,\xi ;\tau ){\int }_{\tau }^{\infty }{\left[Z(\tau ,\xi ;s)\right]}^{-1}{f}_x^0(s,x(\tau ,\xi ;s),u_{*}\left(s\right))ds.$$

(14)

Proof. 

Fix an arbitrary $\tau \ge 0$. According to $\left(A{2}^{\prime }\right)$, for any $\xi \in G_{\tau }$, the process $(x(\tau ,\xi ;·),u_{*}(·))$ satisfies condition $\left(A2\right)$ with the function $\lambda (·)$, which does not depend on $\xi$. Hence, according to ([17], Theorem 2, Theorem 3), the Freshet derivative $W_{\xi }(\tau ,\xi )$ exists for any $\xi \in G_{\tau }$, and it is defined by Formula (14) (see (7)). According to the theorem on the continuous dependence of the solution of the Cauchy problem on the initial data, the matrix function $\xi \mapsto Z(\tau ,\xi ;\tau )$ is continuous on $G_{\tau }$. Analogously, for all $0\le \tau <s$, the matrix function $\xi \mapsto {\left[Z(\tau ,\xi ;s)\right]}^{-1}$ and the scalar function $\xi \mapsto f_x^0(s,x(\tau ,\xi ;s),u_{*}\left(s\right))$ are continuous on $G_{\tau }$. Hence, to complete the proof, it is sufficient to demonstrate that the last improper integral in (14) converges uniformly in $\xi \in G_{\tau }$ ([26], Proposition 5, §17.2.2). According to Lemma 1, this follows immediately from the fact that the function $\lambda (·)$ does not depend on $\xi$. □

Note that if $\xi =x_{*}\left(\tau \right)$, then equality (14) takes the form

$$W_{\xi }(\tau ,x_{*}\left(\tau \right))=Z_{*}\left(\tau \right){\int }_{\tau }^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,$$

where $Z_{*}\left(t\right)\equiv Z(\tau ,x_{*}\left(\tau \right);t)$, $t\ge 0$ is the normalized fundamental matrix solution at time zero to the differential Equation (6).

Main Results

Let $(x_{*}(·),u_{*}(·))$ be a weakly overtaking optimal process in problem $\left(P\right)$, and let the value $J(x_{*}(·),u_{*}(·))$ be finite. Assume that the condition $\left(A{2}^{\prime }\right)$ holds, and for any process $(x(\zeta ;·),u_{*}(·))$ with the initial state $\zeta$ from the $\beta$-neighborhood of the point $x_0$, the condition $\left(A1\right)$ is satisfied. Assume also that $\left(A3\right)$ holds.

Let ${\left\{T_k\right\}}_{k=1}^{\infty }$ be an increasing sequence of positive numbers, such that ${lim}_{k\to \infty }{T}_k=\infty$ and ${lim}_{k\to \infty }\rho (x_{*}\left(T_k\right),M_{\infty })=0$. Since the process $(x_{*}(·),u_{*}(·))$ is admissible, such a sequence ${\left\{T_k\right\}}_{k=1}^{\infty }$ exists (see (4)). Passing to a subsequence, if necessary, we assume that for any $k=1,2,\cdots$, the condition $\rho (x_{*}\left(T_k\right),M_{\infty })\le \epsilon /k$ is satisfied. Here, the number $\epsilon >0$ is defined in $\left(A3\right)$. Let $\delta (T_k,x_{*}\left(T_k\right))>0$ be the number defined in $\left(A3\right)$. By decreasing, if necessary, the number $\delta (T_k,x_{*}\left(T_k\right))>0$, we can assume that for each $k=1,2,\cdots$, the closed $\delta (T_k,x_{*}\left(T_k\right))$-neighborhood of the point $x_{*}\left(T_k\right)$ is contained in the set $G_{T_k}$ (see (11)). Then, for any solution $x(T_k,\xi ;·)$ of the Cauchy problem (5) with the initial condition $x\left(T_k\right)=\xi$, $\parallel \xi -x_{*}\left(T_k\right)\parallel \le \delta (T_k,x_{*}\left(T_k\right))$, we have $\parallel x(T_k,\xi ;0)-x_0\parallel <\beta$.

For $k=1,2,\cdots$, set

$$M_k=\{x\in {\mathbb{R}}^n:\rho (x,M_{\infty })\le \rho (x_{*}\left(T_k\right),M_{\infty }),\parallel x-x_{*}\left(T_k\right)\parallel \le \delta (T_k,x_{*}\left(T_k\right))\}.$$

Then, $M_k$ is a non-empty compact set, $x_{*}\left(T_k\right)\in M_k$, and according to Lemma 3, the function $W_k(·):=W(T_k,·)$ is continuously differentiable on the set $G_{T_k}\supset M_k$.

For $k=1,2,\cdots$, consider the following problem $\left(P_k\right)$ corresponding to the process $(x_{*}(·),u_{*}(·))$ with fixed finite time $T_k$:

$$J_k(x(·),u(·))=\phi \left(x\left(0\right)\right)+W_k\left(x\left(T_k\right)\right)+{\int }_0^{T_k}{f}^0(t,x\left(t\right),u\left(t\right))dt\to \max ,$$

(15)

$$\dot{x}\left(t\right)=f(t,x\left(t\right),u\left(t\right)),$$

(16)

$$x\left(0\right)\in M_0,x\left(T_k\right)\in M_k,$$

(17)

$$u\left(t\right)\in U\left(t\right),$$

(18)

Here, all data are the same as in the original problem $\left(P\right)$.

We consider all Lebesgue measurable functions $u:[0,T_k]\to {\mathbb{R}}^m$ satisfying for any $t\in [0,T_k]$ to inclusion (18) as controls in the problem $\left(P_k\right)$. An admissible trajectory $x(·)$ of a control system (16) corresponding to a control $u(·)$ is an absolutely continuous solution of the differential Equation (16) in G on the interval $[0,T_k]$, which satisfies the endpoint constraints (17), such that the integral in (15) converges. In this case, the pair $\left(x\right(·),u(·\left)\right)$ is called an admissible pair in $\left(P_k\right)$. An admissible pair $(x_k(·),u_k(·))$ is optimal in $\left(P_k\right)$ if the functional (15) reaches its maximum value at this pair over the set of all admissible pairs. Obviously, for any $k=1,2,\cdots$, the restriction of the pair $(x_{*}(·),u_{*}(·))$ to the interval $[0,T_k]$ is an admissible pair in the problem $\left(P_k\right)$.

Lemma 4. 

For any $k=1,2,\cdots$, the restriction of the process $(x_{*}(·),u_{*}(·))$ to the interval $[0,T_k]$ is an optimal admissible pair in $\left(P_k\right)$.

Proof. 

Assume the contrary. Then, there exists a number k and an admissible pair $(x_k(·),u_k(·))$ in $\left(P_k\right)$, such that $J_k(x_k(·),u_k(·))>J_k(x_{*}(·),u_{*}(·))$.

Let us define the composite control $u_{*}^k(·)$ (see (9)) using the equality

$$u_{*}^k\left(t\right)=\left\{\begin{array}{cc}{u}_k\left(t\right),\hfill & t\in [0,T_k),\hfill \\ u_{*}\left(t\right),\hfill & t\in [T_k,\infty ).\hfill \end{array}\right.$$

(19)

Let $x_{*}^k(·)$ be the trajectory corresponding to the control $u_{*}^k(·)$ with the initial state $x_{*}^k\left(0\right)=x_k\left(0\right)\in M_0$. Since $u_{*}^k\left(t\right)\equiv u_k\left(t\right)$ for $t\in [0,T_k)$ (see (19)), we have $x_{*}^k\left(T_k\right)=x_k\left(T_k\right)\in M_k$. According to condition $\left(A3\right)$, the trajectory $x_{*}^k(·)$ is defined on the entire interval $[0,\infty )$, and it is admissible in $\left(P\right)$. Hence, $(x_{*}^k(·),u_{*}^k(·))$ is an admissible process in $\left(P\right)$.

Through construction, we have

$$\begin{array}{c}J(x_{*}^k(·),u_{*}^k(·))=\phi \left(x_k\left(0\right)\right)+W_k\left(x_k\left(T_k\right)\right)+{\int }_0^{T_k}{f}^0(t,x_k\left(t\right),u_k\left(t\right))dt=J_k(x_k(·),u_k(·))\hfill \\ \hfill >J_k(x_{*}(·),u_{*}(·))=\phi \left(x_{*}\left(0\right)\right)+W_k\left(x_{*}\left(T_k\right)\right)+{\int }_0^{T_k}{f}^0(t,x_{*}\left(t\right),u_{*}\left(t\right))dt=J(x_{*}(·),u_{*}(·)).\end{array}$$

However, this contradicts the assumption that $(x_{*}(·),u_{*}(·))$ is a weakly overtaking optimal process in $\left(P\right)$. □

Define the Hamilton–Pontryagin function $\mathcal{H}:[0,\infty )\times G\times U\times {\mathbb{R}}^1\times {\mathbb{R}}^n\to {\mathbb{R}}^1$ and the Hamiltonian $H:[0,\infty )\times G\times {\mathbb{R}}^1\times {\mathbb{R}}^n\to {\mathbb{R}}^1$ of the problem $\left(P\right)$ in a standard way

$$\mathcal{H}(t,x,u,{\psi }^0,\psi )=\langle \psi ,f(t,x,u)\rangle +{\psi }^0{f}^0(t,x,u),$$

$$H(t,x,{\psi }^0,\psi )=\underset{u\in U\left(t\right)}{sup}\mathcal{H}(t,x,u,{\psi }^0,\psi ).$$

In the following, $\widehat{\partial }\phi \left(x\right)$ denotes the limiting subdifferential of a locally Lipschitz function $\phi :G\to {\mathbb{R}}^1$ at a point $x\in G$ ([20], Definition 11.10), ${\widehat{N}}_A\left(a\right)$ denotes the limiting normal cone to a closed set $A\subset {\mathbb{R}}^n$ at the point $a\in A$ [20], and $B^n\left(0\right)=\{x\in {\mathbb{R}}^n:\parallel x\parallel \le 1\}$ denotes the closed unit ball in ${\mathbb{R}}^n$.

The following results provide the necessary optimality conditions of the optimality for the problem $\left(P\right)$ in the form of the Pontryagin maximum principle. Earlier different variants of the maximum principle for problems with asymptotic endpoint constraints were also developed in [13,14,15] but under additional a priori assumptions about the existence of limits of admissible trajectories at infinity.

Theorem 1. 

Let $(x_{*}(·),u_{*}(·))$ be a weakly overtaking optimal process in $\left(P\right)$ for which the improper integral in (1) converges, condition $\left(A{2}^{\prime }\right)$ holds, and for any process $(x(\zeta ;·),u_{*}(·))$ with the initial state ζ from the β-neighborhood of the point $x_0$, the condition $\left(A1\right)$ is satisfied. Assume also that $\left(A3\right)$ holds. Then, there exists a number ${\psi }^0\ge 0$ and a vector $\xi \in {\mathbb{R}}^n$ that do not vanish simultaneously, such that the following conditions hold:

(i) 

The function $\psi :[0,\infty )\to {\mathbb{R}}^n$ defined by the equality

$$\psi \left(t\right)=-Z_{*}\left(t\right)\xi +{\psi }^0{Z}_{*}\left(t\right){\int }_t^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,t\ge 0,$$

(20)

is locally absolutely continuous and satisfies the core conditions of the maximum principle, i.e., $\psi (·)$ is the solution of the adjoint system

$$\dot{\psi }\left(t\right)=-{\mathcal{H}}_x(t,x_{*}\left(t\right),u_{*}\left(t\right),{\psi }^0,\psi \left(t\right)),$$

(21)

and the maximum condition takes place

$$\mathcal{H}(t,x_{*}\left(t\right),u_{*}\left(t\right),{\psi }^0,\psi \left(t\right))\stackrel{a.e.}{=}H(t,x_{*}\left(t\right),{\psi }^0,\psi \left(t\right)).$$

(22)

(ii) 

The following conditions hold:

$$\psi \left(0\right)\in -{\psi }^0\widehat{\partial }\phi \left(x_{*}\left(0\right)\right)+{\widehat{N}}_{M_0}\left(x_{*}\left(0\right)\right),\xi \in \mathrm{Ls}{}_{t\to \infty :\rho (x_{*}\left(t\right),M_{\infty })\to 0}\left\{{\left[Z_{*}\left(t\right)\right]}^{-1}{N}_{*}\left(t\right)\right\}.$$

Here,

$$N_{*}\left(t\right)=\{\lambda \zeta :\lambda \ge 0,\zeta \in \widehat{\partial }\rho (x_{*}\left(t\right),M_{\infty }+\rho (x_{*}\left(t\right),M_{\infty })B^n\left(0\right))\},$$

and the set

$$\begin{array}{c}\mathrm{Ls}{}_{t\to \infty :\rho (x_{*}\left(t\right),M_{\infty })\to 0}\left\{{\left[Z_{*}\left(t\right)\right]}^{-1}{N}_{*}\left(t\right)\right\}=\{\zeta \in {\mathbb{R}}^n:\exists {\left\{t_k\right\}}_{k=1}^{\infty },\underset{k\to \infty }{lim}{t}_k=\infty ,\hfill \\ \hfill \underset{k\to \infty }{lim}\rho (x_{*}\left(t_k\right),M_{\infty })=0,\exists {\left\{{\zeta }_k\right\}}_{k=1}^{\infty },{\zeta }_k\in N_{*}\left(t_k\right):\zeta =\underset{k\to \infty }{lim}{\left[Z_{*}\left(t_k\right)\right]}^{-1}{\zeta }_k\}\end{array}$$

is the upper topological limit of the multivalued mapping $t\mapsto {\left[Z_{*}\left(t\right)\right]}^{-1}{N}_{*}\left(t\right)$ as $t\to \infty$ and $\rho (x_{*}\left(t\right),M_{\infty })\to 0$ ([27], §29).

Proof. 

According to Lemma 4, the restriction of the process $(x_{*}(·),u_{*}(·))$ on $[0,T_k]$ is an optimal pair in $\left(P_k\right)$ for any $k=1,2,\cdots$. Therefore, according to the extended version of the Pontryagin maximum principle for the problem $\left(P_k\right)$ on the fixed time interval $[0,T_k]$, there exists a nonvanishing pair of adjoint variables $({\psi }_k^0,{\psi }_k(·))$, such that the pair $(x_{*}(·),u_{*}(·))$ satisfies the conditions of the Pontryagin maximum principle for the problem $\left(P_k\right)$, together with the pair $({\psi }_k^0,{\psi }_k(·))$ ([20], Theorem 22.26).

This means that ${\psi }_k^0\ge 0$, ${\psi }_k(·)\in AC\left([0,T_k],{\mathbb{R}}^n\right)$, the pair $({\psi }_k^0,{\psi }_k(·))$ is nonvanishing, and the following conditions hold on the time interval $[0,T_k]$:

$${\dot{\psi }}_k\left(t\right)\stackrel{\mathrm{a}.\mathrm{e}}{=}-{\left[f_x(t,x_{*}\left(t\right),u_{*}\left(t\right))\right]}^{*}{\psi }_k\left(t\right)-{\psi }_k^0{f}_x^0(x_{*}\left(t\right),u_{*}\left(t\right)),$$

(23)

$$\mathcal{H}(t,x_{*}\left(t\right),u_{*}\left(t\right),{\psi }_k^0,{\psi }_k\left(t\right))\stackrel{\mathrm{a}.\mathrm{e}.}{=}H(t,x_{*}\left(t\right),{\psi }_k^0,{\psi }_k\left(t\right)),$$

(24)

$${\psi }_k\left(0\right)\in -{\psi }_k^0\widehat{\partial }\phi \left(x_{*}\left(0\right)\right)+{\widehat{N}}_{M_0}\left(x_{*}\left(0\right)\right),{\psi }_k\left(T_k\right)\in {\psi }_k^0\widehat{\partial }{W}_k\left(x_{*}\left(T_k\right)\right)-{\widehat{N}}_{M_k}\left(x_{*}\left(T_k\right)\right).$$

(25)

Since $x_{*}\left(T_k\right)\in \mathrm{int}\{x\in {\mathbb{R}}^n:\parallel x-x_{*}\left(T_k\right)\parallel \le \delta (T_k,x_{*}\left(T_k\right))\}$, we have ${\widehat{N}}_{M_k}\left(x_{*}\left(T_k\right)\right)=\{\lambda \zeta :\lambda \ge 0,\zeta \in \widehat{\partial }\rho (x_{*}\left(T_k\right),M_{\infty }+\rho (x_{*}\left(T_k\right),M_{\infty })B^n\left(0\right))\}$ ([20], Proposition 11.34).

Assume that for each $k=1,2,\cdots$, the function ${\psi }_k(·)$ is extended to the entire interval $[0,\infty )$ by continuity, i.e., ${\psi }_k\left(t\right)\equiv {\psi }_k\left(T_k\right)$, $t\ge T_k$.

According to Lemma 3, the function $W_k(·)$ is continuously differentiable on the open set $G_{T_k}$ containing the point $x_{*}\left(T_k\right)$. Hence, according to Lemma 3 and ([20], Proposition 11.12), we have $\widehat{\partial }W\left(x_{*}\left(T_k\right)\right)=\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)$, and the second inclusion in (25) holds as ${\psi }_k\left(T_k\right)={\psi }_k^0\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)-{\lambda }_k{\zeta }_k$, where ${\lambda }_k\ge 0$, and

$${\zeta }_k\in \widehat{\partial }\rho (x_{*}\left(T_k\right),M_{\infty }+\rho (x_{*}\left(T_k\right),M_{\infty })B^n\left(0\right)).$$

(26)

Using the adjoint system (23) and the Cauchy formula for linear systems ([28], Chapter 4), for any $k=1,2,\cdots$ and arbitrary $t\in [0,T_k]$ we obtain

$$\begin{array}{c}{\psi }_k\left(t\right)=Z_{*}\left(t\right){\left[Z_{*}\left(T_k\right)\right]}^{-1}\left[{\psi }_k^0\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)-{\lambda }_k{\zeta }_k\right]\hfill \\ \hfill +{\psi }_k^0{Z}_{*}\left(t\right){\int }_t^{T_k}{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right)),ds.\end{array}$$

(27)

Recall that the function $Z_{*}(·)$ is a matrix solution of the linear system

$$\dot{z}\left(t\right)=-{\left[f_x(x_{*}\left(t\right),u_{*}\left(t\right))\right]}^{*}z\left(t\right)$$

with the initial condition $Z_{*}\left(0\right)=I$, where I is the unit diagonal $n\times n$ matrix. As the pair $(x_{*}(·),u_{*}(·))$, the matrix function $Z_{*}(·)$ is defined on the entire time interval $[0,\infty )$.

For any $k=1,2,\cdots$, the pair $({\psi }_k^0,{\psi }_k\left(0\right))$ is non-zero. Hence, by multiplying the triple $({\psi }_k^0,{\psi }_k(·),{\lambda }_k)$ by a positive number, without loss of generality, we can assume that the following equality holds:

$${\psi }_k^0+\parallel {\psi }_k\left(0\right)\parallel =1.$$

(28)

Hence, passing if necessary to a subsequence, we obtain ${\psi }_k^0\to {\psi }^0$, ${\psi }_k\left(0\right)\to {\psi }_0$ as $k\to \infty$, where $0\le {\psi }^0\le 1$ and ${\psi }_0\in {\mathbb{R}}^n$. Since the multivalued mappings $a\mapsto {\widehat{N}}_{M_0}\left(a\right)$, $a\in M_0$, and $x\mapsto \widehat{\partial }\phi \left(x\right)$, $x\in G$ are upper semicontinuous, we obtain

$${\psi }_0\in -{\psi }^0\widehat{\partial }\phi \left(x_{*}\left(0\right)\right)+{\widehat{N}}_{M_0}\left(x_{*}\left(0\right)\right).$$

(29)

Using (28), we have

$${\psi }^0+\parallel {\psi }_0\parallel =1.$$

(30)

Consider the sequence of locally absolutely continuous functions ${\left\{{\psi }_k(·)\right\}}_{k=1}^{\infty }$ on $[0,\infty )$. Since for any $k=1,2,\cdots$ the function ${\psi }_k(·)$ is a solution of the adjoint system (23) along the optimal pair $(x_{*}(·),u_{*}(·))$, then according to condition $\left(A1\right)$, the sequence ${\left\{{\psi }_k(·)\right\}}_{k=1}^{\infty }$ of these solutions with the initial states ${\left\{{\psi }_k\left(0\right)\right\}}_{k=1}^{\infty }$ (see (28)) is uniformly bounded and equicontinuous on any finite interval $[0,T]$, $T>0$. Therefore, using the Arzela–Ascoli theorem, passing, if necessary, to a subsequence, without loss of generality, we can assume that there is a locally absolutely continuous function $\psi :[0,\infty )\to {\mathbb{R}}^n$, such that for arbitrary $T>0$, the sequence ${\left\{{\psi }_k(·)\right\}}_{k=1}^{\infty }$ converges to $\psi (·)$ in the space $C([0,T],{\mathbb{R}}^n)$, and, in particular, $\psi \left(0\right)={\psi }_0={lim}_{k\to \infty }{\psi }_k\left(0\right)$. Thus, the pair $({\psi }^0,\psi (·))$ defined in this way is non-zero, and according to (29), the first inclusion in $\left(ii\right)$ holds. Using the condition $\left(A1\right)$ and the relations (23) and (24), it satisfies the adjoint system (21) and the maximum condition (22) on $[0,\infty )$.

Two cases are possible: $\left(a\right)$ There is a sequence of numbers ${\left\{k_i\right\}}_{i=1}^{\infty }$, such that ${\lambda }_{k_i}{\zeta }_{k_i}=0$, $i=1,2,\cdots$; and $\left(b\right)$ For all sufficiently large numbers k, we have ${\lambda }_k\parallel {\zeta }_k\parallel >0$.

Consider case $\left(a\right)$. In this case, passing, if necessary, to a subsequence, we can assume that for all $k=1,2,\cdots$, the equality ${\lambda }_k{\zeta }_k=0$ holds. Then, condition (27) takes the form

$$\begin{array}{c}{\psi }_k\left(t\right)={\psi }_k^0{Z}_{*}\left(t\right){\left[Z_{*}\left(T_k\right)\right]}^{-1}\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\psi }_k^0{Z}_{*}\left(t\right){\int }_t^{T_k}{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,t\ge 0.\end{array}$$

(31)

By virtue of Lemma 3, we have

$$\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)=Z_{*}\left(T_k\right){\int }_{T_k}^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds.$$

Hence, (31) implies the equality

$${\psi }_k\left(t\right)={\psi }_k^0{Z}_{*}\left(t\right){\int }_t^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,t\ge 0.$$

Then, passing to the limit as $k\to \infty$, we obtain

$$\psi \left(t\right)={\psi }^0{Z}_{*}\left(t\right){\int }_t^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,t\ge 0.$$

The pair $({\psi }^0,\psi (·))$ is non-zero, and it satisfies the core conditions of the maximum principle. Hence, without loss of generality, we can put ${\psi }^0=1$. Then, the condition (20) is satisfied with ${\psi }^0=1$ and $\xi =0$ in this case. Note that the equality above coincides with the pointwise representation of the adjoint variable (see (8)) developed in [8,21,24,29] for infinite-horizon problems without asymptotic endpoint constraints under different growth assumptions. Thus, in case $\left(a\right)$, both conditions $\left(i\right)$ and $\left(ii\right)$ hold with the non-zero pair $({\psi }^0,\xi )$, where ${\psi }^0=1$ and $\xi =0$.

Consider case $\left(b\right)$. In this case, without loss of generality, we can assume that for any $k=1,2,\cdots$, we have ${\lambda }_k>0$ and ${\zeta }_k\ne 0$. Since ${\zeta }_k\ne 0$, the point $x_{*}\left(T_k\right)$ belongs to the boundary of the set $M_{\infty }+\rho (x_{*}\left(T_k\right),M_{\infty })B^n\left(0\right)$ (see (26)).

For $k=1,2,\cdots$, we define the vector ${\xi }_k\in {\widehat{N}}_{M_{\infty }+\rho (x_{*}\left(T_k\right),M_{\infty })B^n\left(0\right)}\left(x_{*}\left(T_k\right)\right)$ using the equality

$${\xi }_k={\lambda }_k{\zeta }_k.$$

From (27), we have

$$\begin{array}{c}{\psi }_k\left(0\right)={\left[Z_{*}\left(T_k\right)\right]}^{-1}\left[{\psi }_k^0\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)-{\xi }_k\right]\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\psi }_k^0{\int }_0^{T_k}{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds,\end{array}$$

or, equivalently,

$$\begin{array}{c}{\left[Z_{*}\left(T_k\right)\right]}^{-1}{\xi }_k=-{\psi }_k\left(0\right)+{\psi }_k^0\left\{{\left[Z_{*}\left(T_k\right)\right]}^{-1}\frac{\partial }{\partial x}{W}_k\left(x_{*}\left(T_k\right)\right)+{\int }_0^{T_k}{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(x_{*}\left(s\right),u_{*}\left(s\right))ds\right\}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-{\psi }_k\left(0\right)+{\psi }_k^0\left\{{\int }_{T_k}^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds+{\int }_0^{T_k}{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(x_{*}\left(s\right),u_{*}\left(s\right))ds\right\}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-{\psi }_k\left(0\right)+{\psi }_k^0{\int }_0^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds.\end{array}$$

(32)

Since ${lim}_{k\to \infty }{\psi }_k^0={\psi }^0$ and ${lim}_{k\to \infty }{\psi }_k\left(0\right)={\psi }_0$, the limit on the right-hand side of (32) exists as $k\to \infty$. Hence, the limit on the left-hand side of (32) also exists, and since $\rho (x_{*}\left(T_k\right),M_{\infty })\le \epsilon /k$, we have $\xi ={lim}_{k\to \infty ,}{\left[Z_k\left(T_k\right)\right]}^{-1}{\xi }_k\in \mathrm{Ls}{}_{t\to \infty :\rho (x_{*}\left(t\right),M_{\infty })\to 0}$$\left\{{\left[Z_{*}\left(t\right)\right]}^{-1}{N}_{*}\left(t\right)\right\}$. Thus, the second condition in $\left(ii\right)$ is satisfied for the vector $\xi$.

For an arbitrary $t\ge 0$, passing to the limit in the equality (27) as $k\to \infty$, we obtain

$$\psi \left(t\right)=-Z_{*}\left(t\right)\xi +{\psi }^0{Z}_{*}\left(t\right){\int }_t^{\infty }{\left[Z_{*}\left(s\right)\right]}^{-1}{f}_x^0(s,x_{*}\left(s\right),u_{*}\left(s\right))ds.$$

Moreover, it follows from the equality (30) that the number ${\psi }^0$ and the vector $\xi$ cannot vanish simultaneously. Thus, in the second case, both conditions $\left(i\right)$ and $\left(ii\right)$ are satisfied with the non-zero pair $({\psi }^0,\xi )$. □

In conclusion, we obtain a variant of the maximum principle for the infinite-horizon problem $\left(P\right)$ without asymptotic endpoint constraints.

Corollary 1. 

Assume that there are no asymptotic endpoint constraints in the problem $\left(P\right)$, i.e., $M_{\infty }={\mathbb{R}}^n$ in (4). Let $(x_{*}(·),u_{*}(·))$ be a weakly overtaking optimal process in the problem $\left(P\right)$ for which the functional (1) converges, the condition $\left(A{2}^{\prime }\right)$ holds, and for any process $(x(\zeta ;·),u_{*}(·))$ with the initial state ζ from the β-neighborhood of the point $x_0$, the condition $\left(A1\right)$ is satisfied. Then, the statement of Theorem 1 holds with ${\psi }^0=1$ and $\xi =0$.

Proof. 

Indeed, according to $\left(A{2}^{\prime }\right)$, the condition $\left(A3\right)$ is satisfied automatically in the case of $M_{\infty }={\mathbb{R}}^n$. Further, for any $t\ge 0$, we have $x_{*}\left(t\right)\in \mathrm{int}{M}_{\infty }$ and, consequently, $\widehat{\partial }\rho (x_{*}\left(t\right),M_{\infty }+\rho (x_{*}\left(t\right),M_{\infty })B^n\left(0\right))=0$. Therefore, in this case, the assertion of Theorem 1 holds with $\xi =0$, and without loss of generality, we can assume that ${\psi }^0=1$. □

Note that the variant of the maximum principle in the normal form, similar to the one given in Corollary 1, for the problem $\left(P\right)$ with a fixed initial state and without asymptotic endpoint constraints was obtained earlier in [21] using the needle variations techniques in a more general case when the improper integral in the functional (1) does not necessarily converge, and the less restrictive assumption $\left(A2\right)$ holds instead $\left(A{2}^{\prime }\right)$ (see [9] for details).

4. Example

Here, we consider an application of Theorem 1 to a simple model of the optimal sustainable exploitation of a non-renewable resource.

Assume that the process of extraction of a non-renewable resource is driven by the following control system:

$$\dot{x}\left(t\right)=-u\left(t\right),$$

(33)

$$u\left(t\right)\in (0,\infty ).$$

(34)

Here, $x\left(t\right)\in {\mathbb{R}}^1$ is the resource stock, and $u\left(t\right)\in {\mathbb{R}}^1$ is the amount of the extracted resource at time $t\ge 0$, respectively. Let $x_0>0$ be the initial stock of the resource.

Assume that $a\in (0,x_0)$ is a non-extractable part of the initial stock $x_0$, whose value is determined by a benevolent social planner. An arbitrary measurable function $u:[0,\infty )\to (0,\infty )$ is a control in system (33), (34). The pair $\left(x\right(·),u(·\left)\right)$, where $u(·)$ is a control and $x(·)$ is the corresponding locally absolutely continuous solution to the differential Equation (33), is an admissible process if $x\left(0\right)=x_0$ and ${lim\; sup}_{t\to \infty }x\left(t\right)\ge a$. In this case, any admissible control $u(·)$ is an integrable function, such that ${\int }_0^{\infty }u\left(t\right)dt\le a$.

Assume that the utility of the non-extractable part a of the initial stock $x_0$ is characterized by the value $lna/\rho$, and the utility of an admissible process $\left(x\right(·),u(·\left)\right)$ in (33) and (34) is characterized by the integral functional $J(x(·),u(·))={\int }_0^{\infty }{e}^{-\rho t}lnu\left(t\right)dt$, where $\rho >0$ is a social discount rate. Then, we arrive at the following problem $\left(P1\right)$ of the optimal sustainable exploitation of the non-renewable resource stock $x_0$:

$$J(x(·),u(·),a)={\int }_0^{\infty }{e}^{-\rho t}lnu\left(t\right)dt+\frac{lna}{\rho }\to \max ,$$

(35)

$$\dot{x}\left(t\right)=-u\left(t\right),$$

$$u\left(t\right)\in (0,\infty ),$$

$$x\left(0\right)=x_0,\underset{t\to \infty }{lim\; inf}\rho (x\left(t\right),M_{\infty }\left(a\right))=0,$$

(36)

where $M_{\infty }\left(a\right)=\{x\in {\mathbb{R}}^1:x\ge a\}$. The maximum in the problem $\left(P1\right)$ is sought in all triples $\left(x\right(·),u(·),a)$, where $\left(x\right(·),u(·\left)\right)$ is an admissible process and $a\in (0,x_0)$.

Obviously, to solve the problem $\left(P1\right)$, first, one should find an optimal admissible process $(x_{*}^a(·),u_{*}^a(·))$ in the following problem $\left(Q\right(a\left)\right)$:

$$J(x(·),u(·))={\int }_0^{\infty }{e}^{-\rho t}lnu\left(t\right)dt\to \max ,$$

$$\dot{x}\left(t\right)=-u\left(t\right),$$

$$u\left(t\right)\in (0,\infty ),$$

$$x\left(0\right)=x_0,\underset{t\to \infty }{lim\; inf}\rho (x\left(t\right),M_{\infty }\left(a\right))=0.$$

with a fixed $a\in (0,x_0)$ and the set $G=(0,\infty )$. Then, one should find the value $a_{*}$ that maximizes the function $a\mapsto J(x_{*}^a(·),u_{*}^a(·),a)$ over $(0,x_0)$ (see (35)).

For any fixed $a\in (0,x_0)$, the solution $(x_{*}^a(·),u_{*}^a(·))$ to the problem $\left(Q\right(a\left)\right)$ can be obtained through the application of Theorem 1. Indeed, the problem $\left(Q\right(a\left)\right)$ is a particular case of the problem $\left(P\right)$ with $f^0(t,x,u)\equiv e^{-\rho t}lnu$ and $f(t,x,u)\equiv -u$, $t\ge 0$, $x>0$, $u>0$. It can be directly verified that for any admissible process $(x_{*}(·),u_{*}(·))$, the conditions $\left(A1\right)$, $\left(A{2}^{\prime }\right)$, and $\left(A3\right)$ are satisfied. Thus, if an optimal process $(x_{*}^a(·),u_{*}^a(·))$ exists in the problem $\left(Q\right(a\left)\right)$, then all the conditions of Theorem 1 hold true. Note that for any $a\in (0,x_0)$, for the corresponding conditional cost function, we have $W_x(t,x)\equiv 0$ on $[0,\infty )\times (0,\infty )$.

Therefore, in this case, there are no simultaneously vanishing numbers ${\psi }^0\ge 0$ and $\xi \in {\mathbb{R}}^1$, such that both conditions $\left(i\right)$ and $\left(ii\right)$ of Theorem 1 hold with a function $\psi (·)$ defined by the Cauchy-type formula (20).

In the case of the problem $\left(Q\right(a\left)\right)$, the equality (20) becomes

$$\psi \left(t\right)\equiv \xi ,t\ge 0,$$

where $\xi \ge 0$ is a constant. The maximum condition (22) takes the following form:

$${\psi }^0{e}^{-\rho t}ln{u}_{*}\left(t\right)-\xi u_{*}\left(t\right)\stackrel{\mathrm{a}.\mathrm{e}.}{=}\max {}_{u>0}\left[{\psi }^0{e}^{-\rho t}lnu-\xi u\right].$$

(37)

This implies that $\xi >0$, ${\psi }^0=1$ and $u_{*}\left(t\right)\stackrel{\mathrm{a}.\mathrm{e}.}{=}{\psi }^0{e}^{-\rho t}/\xi$, $t\ge 0$.

Indeed, if $\xi =0$, then ${\psi }^0>0$, and the maximum in (36) cannot be reached for any $t\ge 0$, which contradicts condition $\left(i\right)$ of Theorem 1. Thus, $\xi >0$, and according to (37), we have ${\psi }^0>0$ since otherwise, the maximum in (37) again cannot be reached, which contradicts condition $\left(i\right)$ of Theorem 1.

Thus, without loss of generality, we can assume that ${\psi }^0=1$ and the optimal control $u_{*}(·)$ (if it exists) is defined by the equality

$$u_{*}\left(t\right)\stackrel{\mathrm{a}.\mathrm{e}.}{=}\frac{e^{-\rho t}}{\xi },t\ge 0,$$

where $\xi$ is a positive number. According to (33), the corresponding optimal trajectory $x_{*}(·)$ has the form $x_{*}\left(t\right)=x_0-{\int }_0^t{e}^{-\rho s}ds/\xi$, $t\ge 0$. Hence, due to the optimality of the trajectory $x_{*}(·)$ in the problem $\left(Q\right(a\left)\right)$, we have

$$\underset{t\to \infty }{lim}{x}_{*}\left(t\right)=x_0-\frac{1}{\rho \xi }=a.$$

Therefore, a uniquely sustainable solution for optimality to $\left(Q\right(a\left)\right)$ is the following:

$$\xi =\frac{1}{\rho (x_0-a)},u_{*}^a\left(t\right)\stackrel{\mathrm{a}.\mathrm{e}.}{=}\rho (x_0-a)e^{-\rho t},x_{*}^a\left(t\right)=(x_0-a)e^{-\rho t}+a,t\ge 0.$$

The optimality of the resulting process $(x_{*}^a(·),u_{*}^a(·))$ follows easily from the normality of the problem $\left(Q\right(a\left)\right)$ (${\psi }^0=1$), the concavity of the function $x\mapsto H(t,x,1,\xi )$ on $(0,\infty )$, and the fact that for any $t\ge 0$, the point $(x_{*}\left(t\right),u_{*}\left(t\right))$ is the global maximum point of the function $(x,u)\mapsto \mathcal{H}(t,x,u,1,\xi )$ on the set $(0,\infty )\times (0,\infty )$.

By substituting the optimal control $u_{*}^a(·)$ into (35), we obtain

$$J(x_{*}^a(·),u_{*}^a(·),a)={\int }_0^{\infty }{e}^{-\rho t}\left[ln\rho +ln(x_0-a)-\rho t\right]dt+\frac{lna}{\rho }=\frac{ln\rho +ln(x_0-a)-1}{\rho }+\frac{lna}{\rho }.$$

Finally, by maximizing the value $J(x_{*}^a(·),u_{*}^a(·),a)$ in a over $(0,x_0)$, we obtain the optimal sustainable non-extractable part $a_{*}$ of the initial resource stock $x_0$ and the corresponding optimal process $(x_{*}(·),u_{*}(·))$ in the problem $\left(P1\right)$:

$$a_{*}=\frac{x_0}{2},x_{*}\left(t\right)=\frac{x_0}{2}{e}^{-\rho t}+\frac{x_0}{2},u_{*}\left(t\right)\stackrel{\mathrm{a}.\mathrm{e}.}{=}\frac{\rho x_0}{2}{e}^{-\rho t},t\ge 0.$$

5. Conclusions

In the present paper, a new version of the Pontryagin maximum principle was developed for an infinite-horizon optimal control problem with general asymptotic endpoint constraints. In economics, the fulfillment of asymptotic endpoint constraints of this type can be viewed as the minimal necessary conditions for the sustainability of growth processes. On the basis of the controllability-type assumption, the problem under consideration was reduced to a family of standard finite-horizon problems with utility functionals containing the value of the conditional cost of the phase vector as a terminal term. The method applied was based on exploiting the properties of the conditional cost function. Using this approach, a version of the Pontryagin maximum principle with an explicitly specified adjoint variable was developed. The main novelty of the results is that the general asymptotic endpoint constraints were tackled. Conditions guaranteeing the continuous differentiability of the conditional cost function were presented. A simple model of the optimal sustainable exploitation of a non-renewable resource was considered as an illustrative example. Note that the results obtained here can also be applied to an important problem of the optimal sustainable exploitation of a renewable resource. However, this problem is complex, so it should be the subject of special research.