Pontryagin’s Maximum Principle for Optimal Control Problems Governed by Integral Equations with State and Control Constraints

Abstract

This paper proves a new lemma that characterizes controllability for linear Volterra control systems and shows that the usual controllability assumption for the variational linearized system near an optimal pair is superfluous. Building on this, it establishes a Pontryagin-type maximum principle for Volterra optimal control with general control and state constraints (fixed terminal constraints and time-dependent state bounds), where the cost combines a terminal term with a state-dependent and integral term. Using the Dubovitskii–Milyutin framework, we construct conic approximations for the cost, dynamics, and constraints and derive necessary optimality conditions under mild regularity: (i) a classical adjoint system when only terminal constraints are present and (ii) a Stieltjes-type adjoint with a non-negative Borel measure when pathwise state constraints are active. Furthermore, under convexity of the cost functional and linear Volterra dynamics, the maximum principle becomes a sufficient criterion for global optimality (recovering the classical sufficiency in the differential case). The differential case recovers the classical PMP, and an SIR example illustrates the results. A key theme is symmetry/duality: the adjoint differentiates in the state while the maximum condition differentiates in the control, reflecting operator transposition and the primal–dual geometry of Dubovitskii–Milyutin cones.

1. Introduction

In the past decades, extensive research has been devoted to optimal control problems governed by integral equations. Foundational results for Fredholm-type systems can be found in [1,2] while Volterra-type formulations including nonlinear settings and state–control constraints are treated in [3]. Further contributions appear in [4,5,6]. Applications arise in population dynamics, viscoelastic systems, and epidemic models, often under linear dynamics or simplified kernels. However, many of these works rely on restrictive structural assumptions and do not fully exploit the general Dubovitskii–Milyutin framework.

Our work begins with a new lemma that characterizes the controllability of linear control systems governed by Volterra integral equations. This result, significant in its own right, provides a novel contribution to the theory of integral equations. More importantly, it allows us to eliminate the common assumption regarding the controllability of the variational linear equation around an optimal pair (see Equation (19)).

Several recent contributions have addressed maximum principles for Volterra-type control systems. In [7], systems with unilateral constraints are considered in linear–quadratic frameworks. In [8], nonlinear Volterra equations are studied via discrete-time approximations and modified Hamiltonian methods. A maximum principle for singular kernels and terminal constraints with fractional Caputo dynamics appears in [9]. A rigorous analysis under smooth nonlinearities and mixed control constraints, based on Dubovitskii–Milyutin theory, is presented in [10]. A numerical collocation scheme using Genocchi polynomials for weakly singular kernels is developed in [11].

While each of these studies provides valuable insights, most are limited by restrictive kernel assumptions, simplified dynamics, or narrow cost functionals. By contrast, the present work establishes a general Pontryagin-type maximum principle for nonlinear Volterra systems subject to both terminal and time-dependent state constraints. Building upon our controllability result, we derive necessary optimality conditions using conic approximations within the Dubovitskii–Milyutin framework. It is also worth noting that, beyond the derivation of necessary conditions, we establish sufficient conditions for optimality. In particular, under convexity of the cost functional and linear Volterra dynamics, the maximum principle obtained in Theorem 1 becomes a sufficient criterion for global optimality. This result (Section 5) shows that our framework extends Pontryagin’s maximum principle to integral equations and recovers its classical sufficiency in the differential equation case.

In the case where only terminal constraints are imposed, the adjoint equation simplifies to a modified Volterra integral equation. In the presence of time-dependent state constraints, the adjoint involves a Stieltjes integral with respect to a non-negative Borel measure concentrated on the active set. This structure captures the added complexity of pathwise constraints. Notably, when the Volterra kernel collapses to a differential operator, our results recover the classical Pontryagin maximum principle. Thus, our findings both unify and extend the existing theory for optimal control problems governed by integral and differential systems.

The Dubovitskii–Milyutin method has proved effective across a wide class of optimal control problems, see, for instance, refs. [12,13,14]. In particular, ref. [14] analyzes impulsive control problems without state constraints.

Beyond these theoretical contributions, our use of local cone approximations resonates with developments in numerical analysis and convex approximation methods. Moving asymptotes and local convex surrogates offer computational insights for constrained problems [15], while enrichment techniques in finite element analysis (e.g., truncated Gegenbauer–Hermite weighted approaches) highlight application domains such as viscoelasticity [16].

As a natural extension, the same methodology may be applied to fractional dynamics (see [17,18]), provided that the linear variational equation around an optimal pair is explicitly characterized. Analogous programs apply to integro-dynamic equations on time scales [19] or to problems in spaces of functions of bounded variation [20].

Finally, our results display a natural symmetry between the adjoint equation and the maximum condition. In the adjoint, derivatives act with respect to the state variables, while in the maximum condition they act with respect to the controls. This duality reflects the operator transposition inherent in the Dubovitskii–Milyutin framework, highlighting the structural balance between primal and dual formulations and situating our contribution within symmetry analysis.

2. Problem Statement and Principal Results

Having established the necessary background, we now formulate the optimal control problem studied in this paper. Our aim is to derive the necessary optimality conditions, specifically, a Pontryagin-type maximum principle for systems governed by nonlinear Volterra integral equations in which the state trajectory is subject to both control and state constraints.

We therefore consider the following optimal control problem, which constitutes the core of this work. The objective is to obtain Pontryagin’s necessary conditions for a system whose dynamics follows a nonlinear Volterra integral equation with general constraints on the control and the state, including fixed terminal constraints and time-dependent state bounds. The cost functional contains a terminal term and an integral term that may depend on the state. The general formulation is the following.

Problem 1.

$$\begin{array}{ccc}\hfill h\left(x\right(T\left)\right)& +& {\int }_0^T\phi (x\left(t\right),u\left(t\right),t)dt⟶\mathrm{loc}min,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill (x,u)\in E& =& C^n[0,T]\times L_{\infty }^r[0,T],\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill x\left(t\right)& =& p\left(t\right)+{\int }_0^{t}K(t,s,x\left(s\right),u\left(s\right))ds,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill x\left(T\right)& =& x_1;x_0,x_1\in {\mathbb{R}}^n,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill u\left(t\right)& \in & \Omega ,\mathit{for}\mathit{almost}\mathit{every}t\in [0,T].\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill g\left(x\right(t),t)& \le & 0,t\in [0,T],\hfill \end{array}$$

(6)

where we employ the notation “locmin” to denote a local minimum of the cost functional.

Here, $x\left(t\right)\in {\mathbb{R}}^n$ denotes the state and $u\left(t\right)\in {\mathbb{R}}^r$ the control, which is measurable and takes values in the admissible set $\Omega \subset {\mathbb{R}}^r$. The pair $(x,u)$ lies in the product space $E=C^n[0,T]\times L_{\infty }^r[0,T]$, with ${\mathrm{C}}^n[0,T]=\mathrm{C}([0,T];{\mathbb{R}}^n)$ and $L_{\infty }^r[0,T]=L_{\infty }([0,T];{\mathbb{R}}^r)$. In particular, the state $x(·)$ is continuous, and the control $u(·)$ is essentially bounded. The map $h:{\mathbb{R}}^n\to \mathbb{R}$ represents the terminal cost, $g:{\mathbb{R}}^n\times [0,T]\to \mathbb{R}$ is the state constraint function, and $\phi :{\mathbb{R}}^n\times {\mathbb{R}}^r\times [0,T]\to \mathbb{R}$ is the running cost, depending on $x\left(t\right)$, $u\left(t\right)$, and time t. The function $p:[0,T]\to {\mathbb{R}}^n$ is given, and the nonlinear kernel $K:[0,T]\times [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^r\to {\mathbb{R}}^n$ specifies the Volterra-type dynamics.

We now state the standing assumptions of this paper.

Hypothesis 1.

(a) 

The functions φ, K, h, ${\phi }_x,{\phi }_u,K_x,K_u,K_{x,t},K_{x,t},h_x$ are continuous on compact subsets of ${\mathbb{R}}^n\times {\mathbb{R}}^r\times [0,T]$, and $p\in L_{\infty }^n[0,T]$ with $p\left(0\right)\in {IR}^n$.

(b) 

$\Omega \subset {\mathbb{R}}^r$ is closed and convex, with $int(\Omega )\ne \varnothing$.

(c) 

g is continuous and satisfies $g(x_0,0)<0$ and $g(x_1,T)<0$, where $x_0,x_1\in {\mathbb{R}}^n$ are fixed. Moreover, g has a continuous derivative with respect to its first variable, $g_x$, such that $g_x(x,t)\ne 0$ whenever $g(x,t)=0$.

Now, the more involved version of the maximum principle can be stated. In this setting, the corresponding adjoint equation features a Stieltjes integral with respect to a non-negative Borel measure, thereby accounting for the effect of the state constraints at each time along the trajectory.

Theorem 1.

Assume that conditions (a)–(c) hold for $(x^{\circ },u^{\circ })\in E$, which is a solution to Problem 1.

(α) 

There exists a non-negative Borel measure μ on $[0,T]$ with support contained in

$$R:=\{t\in [0,T]/g(x^{\circ }\left(t\right),t)=0\}\ne \varnothing .$$

(β) 

There exist ${\lambda }_0\in {IR}_{+0}$ and a function $\psi \in L_1^n[0,T]=L_1([0,T];{IR}^n)$ such that ${\lambda }_0$ and ψ do not vanish simultaneously, and ψ is a solution of the adjoint integral equation

$$\begin{array}{cc}\hfill -\psi \left(t\right)=& -{\int }_t^T\left\{{\int }_{\tau }^T{K}_{x,\tau }^{*}\left(s,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)\psi \left(s\right)ds-{\lambda }_0{\phi }_x(x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right),\tau )\right\}d\tau \hfill \\ \hfill & -{\int }_t^T{K}_x^{*}(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(\tau \right)d\tau +{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)\hfill \\ \hfill & -a+{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right),\hfill \end{array}$$

(7)

for some $a\in {IR}^n$. Moreover, for all $U\in \Omega$ and for almost every $t\in [0,T]$ one has

$$〈-{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),U-u\left(t\right)〉\ge 0.$$

Remark 1.

If condition (6) is replaced by the following terminal state constraint:

$$g\left(x\right(T),T)\le 0,$$

(8)

then a simpler version of the maximum principle can be established. In this case, the corresponding adjoint equation does not involve a Stieltjes integral with respect to a non-negative Borel measure, as the state constraint only acts at the terminal time.

Theorem 2.

Assume that conditions (a), (b), and (8) hold and that g has a continuous derivative with respect to its first variable, $g_x$, with $g_x(x^{\circ }\left(T\right),T)\ne 0$. Let $(x^{\circ },u^{\circ })\in E$ be a solution of Problem 1.

Then, there exist ${\lambda }_0\ge 0$, ${\mu }_0\ge 0$ and a function $\psi \in C^1[0,T;{\mathbb{R}}^n]$ such that ${\lambda }_0$ and ψ do not vanish simultaneously, and ψ solves the system

$$\begin{array}{cc}\hfill {\psi }^{\prime }\left(t\right)=& -{\int }_t^T{K}_{x,t}^{*}\left(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\right)\psi \left(s\right)ds+{\lambda }_0{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)\hfill \\ \hfill & -K_x^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\mu }_0{g}_x(x^{\circ }\left(T\right),T),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \psi \left(T\right)=& a-{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right),\hfill \end{array}$$

(10)

and, moreover, for all $U\in \Omega$ and for almost every $t\in [0,T]$ one has

$$〈-{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),U-u\left(t\right)〉\ge 0.$$

Corollary 1.

Assume that hypotheses (a)–(c) hold for $(x^{\circ },u^{\circ })\in E$, a solution to Problem 1, and that

$$K_{x,\tau }\left(s,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)=0,\forall s,\tau \in [0,T].$$

(11)

(α) 

There exists a non-negative Borel measure μ on $[0,T]$ whose support is contained in

$$R:=\{t\in [0,T]:g(x^{\circ }\left(t\right),t)=0\}\ne \varnothing .$$

(β) 

There exist ${\lambda }_0\in {IR}_{+0}$ and a function $\psi \in L_1^n[0,T]=L_1([0,T];{IR}^n)$ such that ${\lambda }_0$ and ψ do not vanish simultaneously, and ψ solves the integral equation

$$\begin{array}{cc}\hfill -\psi \left(t\right)=& -{\int }_t^T{K}_x^{*}(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(\tau \right)d\tau +{\int }_t^T{\lambda }_0{\phi }_x(x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right),\tau )d\tau \hfill \\ \hfill & +{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)-a+{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right),\hfill \end{array}$$

(12)

for some $a\in {IR}^n$. Moreover, for all $U\in \Omega$ and for almost every $t\in [0,T]$ one has

$$〈-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),U-u\left(t\right)〉\ge 0.$$

3. Preliminary Results

This section begins with a lemma that provides a characterization of the controllability of linear control systems governed by Volterra integral equations. The lemma is of independent interest and constitutes a novel contribution to the theory of integral equations. It also enables us to show that the commonly imposed controllability of the variational linear equation around an optimal pair is, in fact, redundant.

We then give a concise overview of the Dubovitskii–Milyutin (DM) framework. We formulate the constrained optimization problem and construct approximation cones for both the objective and the constraints. The optimality condition, written in an Euler–Lagrange (EL) form, is expressed via the corresponding dual cones. These statements rely on standard principles of DM theory; complete proofs can be found in [13,14,21].

3.1. Controllability of a Linear Volterra Equation

In this subsection, we state a lemma that characterizes the controllability of the linear control system

$$x\left(t\right)={\int }_0^t\left[A(t,s)x\left(s\right)+B(t,s)u\left(s\right)\right]ds,$$

(13)

where $x\left(t\right)\in {\mathbb{R}}^n$ is the state and $u\left(t\right)\in {\mathbb{R}}^r$ is the control. The matrix $A(t,s)$ is $n\times n$ and $B(t,s)$ is $n\times r$, with all entries in $C^1(\Delta ;\mathbb{R})$, where $\Delta =\{(t,s)\in {[0,T]}^2:0\le s\le t\le T\}$. In particular, A and B are continuously differentiable in both variables on the triangular domain $\Delta$.

Lemma 1.

System (13) is not controllable if and only if there exists a nonzero function $\psi \in C([0,T],{\mathbb{R}}^n)$ such that

$${\psi }^{\prime }\left(t\right)=-{\int }_t^T{\displaystyle \frac{\partial A^{*}}{\partial t}}(s,t)\psi \left(s\right)ds-A^{*}(t,t)\psi \left(t\right),t\in [0,T],$$

(14)

and

$${\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds+B^{*}(t,t)\psi \left(t\right)=0$$

(15)

for all $t\in [0,T]$.

Proof. 

Assume first that (13) is not controllable. Then, the set

$$D=\left\{x\left(T\right):x\mathrm{solves}\left(13\right)\right\}\subset {\mathbb{R}}^n$$

is a proper linear subspace; that is, $D\ne {\mathbb{R}}^n$. Hence, there exists $a\in {\mathbb{R}}^n$, $a\ne 0$, such that

$$\langle a,x\left(T\right)\rangle =0,\forall x\left(T\right)\in D.$$

Let $\psi$ be the solution of the equation

$$\left\{\begin{array}{c}{\displaystyle {\psi }^{\prime }\left(t\right)=-{\int }_0^T\frac{\partial A^{*}}{\partial t}(s,t)\psi \left(s\right)ds-A^{*}(t,t)\psi \left(t\right),t\in [0,T].}\hfill \\ \psi \left(T\right)=a.\hfill \end{array}\right.$$

(16)

Fix any $u\in L^{\infty }([0,T];{\mathbb{R}}^r)$. Let $x(·)$ be the corresponding solution of the Equation (13). Then,

$$\begin{array}{cc}\hfill {\int }_0^T\langle x\left(t\right),{\psi }^{\prime }\left(t\right)\rangle dt& ={\int }_0^T〈x\left(t\right),-{\int }_t^T{\displaystyle \frac{\partial A^{*}}{\partial t}}(s,t)\psi \left(s\right)ds〉dt-{\int }_0^T\langle x\left(t\right),A^{*}(t,t)\psi \left(t\right)\rangle dt\hfill \\ \hfill & =-{\int }_0^T{\int }_t^T\langle x\left(t\right),\frac{\partial A^{*}}{\partial t}(s,t)\psi \left(s\right)\rangle dsdt-{\int }_0^T\langle A(t,t)x\left(t\right),\psi \left(t\right)\rangle dt\hfill \\ \hfill & =-{\int }_0^T〈{\int }_0^t\frac{\partial A}{\partial t}(t,s)x\left(s\right)ds,\psi \left(t\right)〉dt-{\int }_0^T\langle A(t,t)x\left(t\right),\psi \left(t\right)\rangle dt\hfill \\ \hfill & =-{\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}A(t,s)x\left(s\right)ds\right),\psi \left(t\right)〉dt.\hfill \end{array}$$

In addition,

$$\begin{array}{ccc}\hfill {\int }_0^T\langle x^{\prime }\left(t\right),\psi \left(t\right)\rangle dt& =& {\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}A(t,s)x\left(s\right)ds\right),\psi \left(t\right)〉dt\hfill \\ & +& {\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}B(t,s)u\left(s\right)ds\right),\psi \left(t\right)〉dt.\hfill \end{array}$$

Integrating by parts and using $\psi \left(T\right)=a$, we obtain

$$\begin{array}{cc}\hfill {\int }_0^T\langle x^{\prime }\left(t\right),\psi \left(t\right)\rangle dt& =\psi \left(T\right)x\left(T\right)-\psi \left(0\right)x\left(0\right)-{\int }_0^T\langle {\psi }^{\prime }\left(t\right),x\left(t\right)\rangle dt\hfill \\ \hfill & ={\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}A(t,s)x\left(s\right)ds\right),\psi \left(t\right)〉dt.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill 0& ={\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}B(t,s)u\left(s\right)ds\right),\psi \left(t\right)〉dt\hfill \\ \hfill & ={\int }_0^T〈{\int }_0^t\frac{\partial B}{\partial t}(t,s)u\left(s\right)ds,\psi \left(t\right)〉dt+{\int }_0^T\langle B(t,t)u\left(t\right),\psi \left(t\right)\rangle dt\hfill \\ \hfill & ={\int }_0^T〈{\int }_t^T\frac{\partial B^{*}}{\partial t}(s,t)\psi \left(s\right)ds,u\left(t\right)〉dt+{\int }_0^T\langle B^{*}(t,t)\psi \left(t\right),u\left(t\right)\rangle dt\hfill \\ \hfill & ={\int }_0^T〈{\int }_t^T\frac{\partial B^{*}}{\partial t}(s,t)\psi \left(s\right)ds+B^{*}(t,t)\psi \left(t\right),u\left(t\right)〉dt,\hfill \end{array}$$

for every $u\in L_{\infty }^r[0,T]$. Therefore,

$${\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds+B^{*}(t,t)\psi \left(t\right)=0,t\in [0,T].$$

Conversely, assume that there exists $\psi$ with $\psi \left(T\right)\ne 0$ satisfying (14) and (15). For any solution x of (13), repeating the steps above and integrating by parts yields

$$\begin{array}{cc}\hfill 0& ={\int }_0^T〈{\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds+B^{*}(t,t)\psi \left(t\right),u\left(t\right)〉dt\hfill \\ \hfill & ={\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}B(t,s)u\left(s\right)ds\right),\psi \left(t\right)〉dt\hfill \\ \hfill & ={\int }_0^T\langle x^{\prime }\left(t\right),\psi \left(t\right)\rangle dt-{\int }_0^T〈{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^{t}A(t,s)x\left(s\right)ds\right),\psi \left(t\right)〉dt\hfill \\ \hfill & =\psi \left(T\right)x\left(T\right)-\psi \left(0\right)x\left(0\right)=\psi \left(T\right)x\left(T\right).\hfill \end{array}$$

Thus, D admits $\psi \left(T\right)\ne 0$ as a perpendicular vector, so D is not dense in ${\mathbb{R}}^n$; i.e., $D\ne {\mathbb{R}}^n$. Hence, system (13) is not controllable. □

3.2. DM Theorem

Let E be a locally convex topological vector space and let $E^{*}$ denote its space of continuous linear functionals.

A subset $\bigwedge \subset E$ will be called a cone with vertex at the origin if it is closed under positive scalings; i.e., $\beta \bigwedge =\bigwedge$ for every $\beta >0$.

The dual cone associated with ⋀ is

$${\bigwedge }^{+}=\{\mathfrak{L}\in E^{*}:\mathfrak{L}\left(x\right)\ge 0\mathrm{for}\mathrm{all}x\in \bigwedge \}.$$

We recall a collection of standard consequences from the Dubovitskii–Milyutin framework (see [21]). First, given a family $\{{\bigwedge }_{\alpha }\subset E:\alpha \in \mathfrak{A}\}$ of convex cones that are weakly closed,

$${\left(\bigcap _{\alpha \in \mathfrak{A}}{\bigwedge }_{\alpha }\right)}^{+}=\overline{\sum _{\alpha \in \mathfrak{A}}{\bigwedge }_{\alpha }^{+}},$$

where the bar denotes closure in the $w^{*}$-topology. This identity reflects how separation in the dual space converts intersections into (closed) sums.

Next, if ${\bigwedge }_1,\dots ,{\bigwedge }_n\subset E$ are open convex cones with a nonempty common part,

$$\bigcap _{i=1}^n{\bigwedge }_i\ne \varnothing ,$$

then the dual of their intersection decomposes additively:

$${\left(\bigcap _{i=1}^n{\bigwedge }_i\right)}^{+}=\sum _{i=1}^n{\bigwedge }_i^{+}.$$

In other words, under an interiority condition, dualization turns intersections into (finite) sums without requiring additional closure.

Finally, consider convex cones ${\bigwedge }_1,\dots ,{\bigwedge }_{n+1}\subset E$ with vertex at the origin and assume that ${\bigwedge }_1,\dots ,{\bigwedge }_n$ are open. Then, the geometric alternative

$$\bigcap _{i=1}^{n+1}{\bigwedge }_i=\varnothing ,$$

holds precisely when there exist functionals ${\mathfrak{L}}_i\in {\bigwedge }_i^{+}$ for $i=1,\dots ,n+1$, not all zero, such that their sum cancels:

$${\mathfrak{L}}_1+{\mathfrak{L}}_2+{\mathfrak{L}}_3+\dots +{\mathfrak{L}}_n+{\mathfrak{L}}_{n+1}=0.$$

This is a dual separation statement for cones: emptiness of the primal intersection is certified by a nontrivial balanced combination of supporting functionals from the corresponding dual cones.

3.3. Euler–Lagrange (EL) Framework

Let $\mathfrak{L}:E\to \mathbb{R}$ be a functional and let ${\mho }_i\subset E$ for $i=1,2,3,4,\dots ,n+1$, with $int\left({\mho }_i\right)\ne \varnothing$ for $i=1,\dots ,n$. We consider the optimization task

$$\left\{\begin{array}{c}\mathfrak{L}\left(x\right)⟶\mathrm{loc}min\hfill \\ \\ x\in {\mho }_i,i=1,2,3,4,\dots ,n+1.\hfill \end{array}\right.$$

(17)

Remark 2.

In customary settings, the families ${\mho }_i$ for $i=1,\dots ,n$ encode inequality-type admissibility regions, whereas ${\mho }_{n+1}$ represents equality-type restrictions; typically $int\left({\mho }_{n+1}\right)=\varnothing$.

Theorem 3.

(DM). Let $x^{\circ }\in E$ be a (local) minimizer of (17) and suppose the following:

(a) 

${\bigwedge }_0$ denotes the decay (descent) cone of $\mathfrak{L}$ at $x^{\circ }$.

(b) 

For $i=1,2,\dots ,n$, the sets ${\bigwedge }_i$ are the feasible direction cones attached to ${\mho }_i$ at the point $x^{\circ }\in {\mho }_i$.

(c) 

${\bigwedge }_{n+1}$ is the tangent cone to ${\mho }_{n+1}$ at $x^{\circ }$.

If all the cones ${\bigwedge }_i$, $i=0,1,\dots ,n+1$, are convex, then there exist continuous linear functionals ${\mathfrak{L}}_i\in {\bigwedge }_i^{+}$ for $i=0,1,\dots ,n+1$, not all identically zero, such that

$${\mathfrak{L}}_0+{\mathfrak{L}}_1+dots+{\mathfrak{L}}_{n+1}=0.$$

(18)

Remark 3.

The relation (18) is often referred to as the abstract Euler–Lagrange (EL) identity. Definitions of the decay/descent, feasible direction, and tangent cones can be found in [21].

Scheme to apply the Dubovitskii–Milyutin Theorem to specific problems:

(i) 

Identify the decay vectors.

(ii) 

Specify the admissible vectors.

(iii) 

Characterize the tangent vectors.

(iv) 

Construct the corresponding dual cones.

We denote by ${\bigwedge }_d={\bigwedge }_d(\mathfrak{L},x^{\circ })$ the decay/descent cone and obtain the following important result.

Theorem 4

(see [21], p. 45). Let $\mathfrak{L}:E\to \mathbb{R}$ be convex and continuous on a topological vector space E, and let $x^{\circ }\in E$. Then, the (one-sided) directional derivative exists in every direction at $x^{\circ }$, and

(a) 

${\displaystyle {\mathfrak{L}}^{\prime }(x^{\circ },h)=inf\left\{\frac{\mathfrak{L}(x^{\circ }+\epsilon h)-\mathfrak{L}\left(x^{\circ }\right)}{\epsilon }:\epsilon \in {\mathbb{R}}_{+}\right\};}$

(b) 

${\displaystyle {\bigwedge }_d(\mathfrak{L},x^{\circ })=\{h\in E:{\mathfrak{L}}^{\prime }(x^{\circ },h)<0\}.}$

We write ${\bigwedge }_a={\bigwedge }_a(\mho ,x^{\circ })$ for the admissible (feasible direction) cone of ℧ at $x^{\circ }\in \mho$.

Theorem 5

(see [21], p. 59). If $\mho \subset E$ is convex with $int(\mho )\ne \varnothing$, then

$${\bigwedge }_a=\left\{h\in E:h=\lambda (x-x^{\circ }),x\in int(\mho ),\lambda \in {\mathbb{R}}_{+}\right\}.$$

The collection of all angent directions to ℧ at $x^{\circ }$ forms a cone with vertex at the origin; we denote it by ${\bigwedge }_T:={\bigwedge }_T(\mho ,x^{\circ })$ and call it the angent cone.

A key device for evaluating tangent cones is the Lyusternik principle, which reduces the computation to a linearized condition at the reference point.

Theorem 6

(Lyusternik; see [21]). Let $E_1,E_2$ be Banach spaces and suppose the following:

(a) 

$x^{\circ }\in E_1$ and $P:E_1\to E_2$ is Fréchet-differentiable at $x^{\circ }$;

(b) 

The derivative $P^{\prime }\left(x^{\circ }\right):E_1\to E_2$ is onto.

Then, for the set $\mho :=\{x\in E_1:P\left(x\right)=0\}$, the tangent cone at $x^{\circ }\in \mho$ is

$${\bigwedge }_T=ker{P}^{\prime }\left(x^{\circ }\right).$$

A detailed proof of this Theorem can be found in [22] (also, see [23]).

4. Establishing the Main Result Theorem 1

Proof. 

Define the functional $\overline{\mathfrak{L}}:E\to \mathbb{R}$ by

$$\begin{array}{c}\hfill \overline{\mathfrak{L}}(x,u)=h\left(x\left(T\right)\right)+{\int }_0^T\phi (x\left(t\right),u\left(t\right),t)dt,\end{array}$$

and set $\mho :={\mho }_1\cap {\mho }_2\cap {\mho }_3$, where ${\mho }_1,{\mho }_2,{\mho }_3$ collect the pairs $(x,u)\in E$ that satisfy, respectively, (3), (4), (5), and (6).

With this notation, Problem 1 can be reformulated as

$$\left\{\begin{array}{c}\overline{\mathfrak{L}}(x,u)⟶\mathrm{loc}min,\hfill \\ \\ (x,u)\in \mho .\hfill \end{array}\right.$$

(a)

Step (a): Behavior of $\overline{\mathfrak{L}}$.

Let ${\bigwedge }_0:={\bigwedge }_d\left(\overline{\mathfrak{L}},(x^{\circ },u^{\circ })\right)$ denote the descent cone of $\overline{\mathfrak{L}}$ at $(x^{\circ },u^{\circ })$. By results presented in the preliminaries and in [12,13,14], we have that

$${\bigwedge }_0=\left\{(x,u)\in E:{\overline{\mathfrak{L}}}^{\prime }(x^{\circ },u^{\circ })(x,u)<0\right\}.$$

Whenever ${\bigwedge }_0\ne ⌀$, its dual cone is

$${\bigwedge }_0^{+}=\left\{-{\lambda }_0{\overline{\mathfrak{L}}}^{\prime }(x^{\circ },u^{\circ }):{\lambda }_0\ge 0\right\}.$$

Moreover, as in [21], we have that ${\overline{\mathfrak{L}}}^{\prime }(x^{\circ },u^{\circ })$ is given, for all $(x,u)\in E.$, as follows:

$${\overline{\mathfrak{L}}}^{\prime }(x^{\circ },u^{\circ })(x,u)=\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\int }_0^T\left[{\phi }_x(x^{\circ },u^{\circ },t)x\left(t\right)+{\phi }_u(x^{\circ },u^{\circ },t)u\left(t\right)\right]dt.$$

Consequently, for any ${\mathfrak{L}}_0\in {\bigwedge }_0^{+}$ there exists ${\lambda }_0\ge 0$ such that, for all $(x,u)\in E$,

$${\mathfrak{L}}_0(x,u)=-{\lambda }_0\left\{\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\int }_0^T\left[{\phi }_x(x^{\circ },u^{\circ },t)x\left(t\right)+{\phi }_u(x^{\circ },u^{\circ },t)u\left(t\right)\right]dt\right\}.$$

(see [12,13,14]).

(b)

Examination of the constraint ${\mho }_1.$

We now compute the tangent cone to ${\mho }_1$ at the reference point $(x^{\circ },u^{\circ })$:

$${\bigwedge }_1:={\bigwedge }_T\left({\mho }_1,(x^{\circ },u^{\circ })\right).$$

Assume, for the moment, that the following variational integral system

$$y\left(t\right)={\int }_0^t\left[K_x\left(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right)\right)y\left(s\right)+K_u\left(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right)\right)u\left(s\right)\right]ds,$$

(19)

is controllable. Invoking Theorem 6 (see [12,13,14,21]), one obtains

$$\begin{array}{cc}\hfill {\bigwedge }_1& =\{(x,u)\in E:x\left(t\right)={\int }_0^t\left[K_x\left(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right)\right)x\left(s\right)+K_u\left(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right)\right)u\left(s\right)\right]ds,\hfill \\ & x\left(T\right)=0,t\in [0,T]\}.\hfill \end{array}$$

To determine ${\bigwedge }_1^{+}$, consider the linear subspaces

$$\begin{array}{cc}\hfill L_1& :=\left\{(x,u)\in E:x\left(t\right)={\int }_0^t\left[K_x\left(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right)\right)x\left(s\right)+K_u\left(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right)\right)u\left(s\right)\right]ds\right\},\hfill \\ \hfill L_2& :=\left\{(x,u)\in E:x\left(T\right)=0\right\}.\hfill \end{array}$$

Hence ${\bigwedge }_1=L_1\cap L_2$. By Proposition 2.40 in [14] (see also [13]), we have that: a functional ${\mathfrak{L}}_{12}$ belongs to $L_2^{+}$ if there exists $a\in {\mathbb{R}}^n$ such that

$${\mathfrak{L}}_{12}(x,u)=\langle a,x\left(T\right)\rangle ,(x,u)\in E.$$

Moreover, Lemma 2.5 in [14] ensures that $L_1^{+}+L_2^{+}$ is $w^{*}$-closed; using basic properties of dual cones, it follows that

$${\bigwedge }_1^{+}=L_1^{+}+L_2^{+}.$$

Therefore, a functional ${\mathfrak{L}}_1$ lies in ${\bigwedge }_1^{+}$ precisely when it can be decomposed as ${\mathfrak{L}}_1={\mathfrak{L}}_{11}+{\mathfrak{L}}_{12}$ with ${\mathfrak{L}}_{11}\in L_1^{+}$ and ${\mathfrak{L}}_{12}\in L_2^{+}$.

(c)

Handling of the constraint ${\mho }_2.$

Introduce

$${\mho }_2^{\prime }:=\left\{u\in L_{\infty }^r[0,T]:u\left(t\right)\in \Omega \mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,T]\right\}.$$

With this notation, ${\mho }_2=C^n[0,T]\times {\mho }_2^{\prime }$. Because $\Omega$ is closed, convex, and has a nonempty interior, we immediately obtain the following:

• ${\mho }_2$ and ${\mho }_2^{\prime }$ are convex and closed (in the respective ambient topologies);
• $int\left({\mho }_2\right)\ne \varnothing$ and $int\left({\mho }_2^{\prime }\right)\ne \varnothing$.

Let ${\bigwedge }_2$ denote the cone of feasible directions for ${\mho }_2$ at the point $(x^{\circ },u^{\circ })\in {\mho }_2$. Then, the product structure carries over to cones:

$${\bigwedge }_2=C^n[0,T]\times {\bigwedge }_2^{\prime },$$

where ${\bigwedge }_2^{\prime }$ is the feasible direction cone of ${\mho }_2^{\prime }$ at $u^{\circ }\in {\mho }_2^{\prime }$.

Consequently, any supporting functional ${\mathfrak{L}}_2\in {\bigwedge }_2^{+}$ splits as

$${\mathfrak{L}}_2=(0,{\mathfrak{L}}_2^{\prime })\mathrm{for}\mathrm{some}{\mathfrak{L}}_2^{\prime }\in {\bigwedge }_2^{\prime +}.$$

By Theorem 5, the functional ${\mathfrak{L}}_2^{\prime }$ acts as a support to the set ${\mho }_2^{\prime }$ at the reference control $u^{\circ }$.

(d)

Treatment of the constraint ${\mho }_3.$

Introduce the scalar functional

$$\begin{array}{ccc}\hfill l& :& C^n[0,T]⟶\mathbb{R},\hfill \\ \hfill l\left(x\right)& =& {\displaystyle \underset{t\in [0,T]}{max}g\left(x\left(t\right),t\right).}\hfill \end{array}$$

By Example 7.5 from [21], its directional derivative at $x^{\circ }$ in the direction $h\in C^n[0,T]$ is

$$l^{\prime }(x^{\circ };h)=\underset{t\in R}{max}〈g_x\left(x^{\circ }\left(t\right),t\right),h\left(t\right)〉,$$

where

$$R=\left\{t\in [0,T]:g\left(x^{\circ }\left(t\right),t\right)=l\left(x^{\circ }\right)\right\}.$$

Since along the active pathwise constraint, we have $l\left(x^{\circ }\right)=0$, this set reduces to

$$R=\left\{t\in [0,T]:g\left(x^{\circ }\left(t\right),t\right)=0\right\}.$$

On the other hand, the feasible direction cone for ${\mho }_3$ at $(x^{\circ },u^{\circ })$ contains the descent cone:

$${\bigwedge }_3={\bigwedge }_a\left({\mho }_3,(x^{\circ },u^{\circ })\right)\supset {\bigwedge }_d\left(\mathfrak{L},(x^{\circ },u^{\circ })\right)=:{\bigwedge }_d.$$

(20)

By Theorem 4, this latter cone admits the representation

$$\begin{array}{ccc}\hfill {\bigwedge }_d& =& \left\{(h,u)\in E:l^{\prime }(x^{\circ };h)<0\right\}\hfill \\ & =& \left\{(h,u)\in E:\langle g_x(x^{\circ }\left(t\right),t),h\left(t\right)\rangle <0\mathrm{for}\mathrm{all}t\in R\right\}.\hfill \end{array}$$

Consequently, from (20), we obtain the inclusion of dual cones

$${\bigwedge }_3^{+}\subset {\bigwedge }_d^{+}.$$

Then, by Example 10.3 (see [21], p. 73), we have that for all $\mathfrak{L}\in {\bigwedge }_3^{+},$ there is a non-negative Borel measure $\mu$ on $[0,T]$ such that

$$\mathfrak{L}(x,u)=-{\int }_0^T{g}_x(x^{\circ }\left(t\right),t)x\left(t\right)d\mu \left(t\right),((x,u)\in E),$$

and $\mu$ has support in

$$R=\{t\in [0,T]/g(x^{\circ }\left(t\right),t)=0\}.$$

(e)

Euler–Lagrange relation.

Note first that each of the sets ${\bigwedge }_0,{\bigwedge }_1,{\bigwedge }_2,{\bigwedge }_3$ is a convex cone. Consequently, by Theorem 3 there exist functionals ${\mathfrak{L}}_i\in {\bigwedge }_i^{+}$ for $i=0,1,2,3$, not all trivial, such that

$${\mathfrak{L}}_0+{\mathfrak{L}}_1+{\mathfrak{L}}_2+{\mathfrak{L}}_3=0.$$

(21)

Expanding (21) using the representations introduced earlier, we obtain

$$\begin{array}{cc}& -{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(t\right)-{\lambda }_0{\int }_0^T\left[{\phi }_x(x^{\circ },u^{\circ },t)x\left(t\right)+{\phi }_u(x^{\circ },u^{\circ },t)u\left(t\right)\right]dt\hfill \\ & +{\mathfrak{L}}_{11}(x,u)+\langle a,x\left(T\right)\rangle +{\mathfrak{L}}_3(x,u)+{\mathfrak{L}}_2^{\prime }\left(u\right)=0,((x,u)\in E).\hfill \end{array}$$

Next, given any $u\in L_{\infty }^r$, there exists $x\in C^n[0,T]$ solving (19) with initial condition $x\left(0\right)=0$. Hence $(x,u)\in L_1$ and, in particular, ${\mathfrak{L}}_{11}(x,u)=0$. Therefore the previous identity reduces to

$$\begin{array}{cc}\hfill {\mathfrak{L}}_2^{\prime }\left(u\right)=& {\lambda }_0{\int }_0^T{\phi }_x(x^{\circ },u^{\circ },t)x\left(t\right)dt+{\lambda }_0{\int }_0^T{\phi }_u(x^{\circ },u^{\circ },t)u\left(t\right)dt\hfill \\ \hfill & -\langle a,x\left(t\right)\rangle +{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\int }_0^T\langle g_x(x^{\circ }\left(t\right),t),x\left(t\right)\rangle d\mu \left(t\right).\hfill \end{array}$$

(22)

Let $\psi$ denote the solution of the adjoint relation (12); that is,

$$\begin{array}{cc}\hfill -\psi \left(t\right)=& -{\int }_t^T\left\{{\int }_{\tau }^T{K}_{x,\tau }^{*}\left(s,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)\psi \left(s\right)ds-{\lambda }_0{\phi }_x\left(x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right),\tau \right)\right\}d\tau \hfill \\ \hfill & -{\int }_t^T{K}_x^{*}\left(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)\psi \left(\tau \right)d\tau +{\int }_t^T{g}_x\left(x^{\circ }\left(\tau \right),\tau \right)d\mu \left(\tau \right)\hfill \\ \hfill & -a+{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right).\hfill \end{array}$$

This is a Volterra-type integral equation of the second kind, which admits a unique solution $\psi \in L_1^n[0,T]$ (see [24], p. 519). Consequently, multiplying both sides by $x\left(t\right)$ and integrating with respect to t over the interval $[0,T]$, we obtain

$$\begin{array}{cc}\hfill -{\int }_0^T\langle \psi \left(t\right),x^{\prime }\left(t\right)\rangle dt& =-{\int }_0^T\left({\int }_t^T\langle {\int }_{\tau }^T{K}_{x\tau }^{*}(s,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(s\right)ds,x^{\prime }\left(t\right)\rangle d\tau \right)dt\hfill \\ \hfill & -{\int }_0^T〈{\int }_t^T{K}_x^{*}(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(\tau \right)d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ \hfill & -{\lambda }_0{\int }_0^T\ge 〈{\int }_t^T{\phi }_x(x^{\circ }\left(\tau \right),{\mu }^{\circ }\left(\tau \right),\tau )d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ \hfill & +{\int }_0^T〈{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )\mu \left(\tau \right)d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ \hfill & -\langle a,x\left(T\right)\rangle +{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)\hfill \end{array}$$

(23)

By integrating by parts in t and exchanging the order of integration, the first contribution in (23) can be rewritten as follows:

$$\begin{array}{cc}\hfill & -{\int }_0^T\left({\int }_t^T〈{\int }_{\tau }^T{K}_{x\tau }^{*}\left(s,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)\psi \left(s\right)ds,x^{\prime }\left(t\right)〉d\tau ,\psi \left(t\right)\right)dt\hfill \\ \hfill & =-{\int }_0^T\left({\int }_t^T{K}_{xt}^{*}\left(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\right)\psi \left(s\right)ds,x\left(t\right)\right)dt\hfill \\ \hfill & =-{\int }_0^T\left({\int }_0^t{K}_{xt}\left(t,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)x\left(\tau \right)d\tau ,\psi \left(t\right)\right)dt.\hfill \end{array}$$

(24)

Similarly, the second term of (23) can be written as follows:

$$\begin{array}{ccc}\hfill & & -{\int }_0^T〈{\int }_t^T{K}_x^{*}(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(\tau \right)d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ & =& {\int }_0^T〈{\int }_T^t{K}_x^{*}(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(\tau \right)d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ & =& -{\int }_0^T\langle K_x^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(t\right),x\left(t\right)\rangle dt\hfill \\ & =& -{\int }_0^T〈\psi \left(t\right),K_x(t,t,x^{\circ }\left(\tau \right),u^{\circ }\left(t\right)x\left(t\right)〉dt.\hfill \end{array}$$

(25)

Likewise, we have that for the third term

$$\begin{array}{cc}\hfill & -{\lambda }_0{\int }_0^T\ge 〈{\int }_t^T{\phi }_x(x^{\circ }\left(\tau \right),{\mu }^{\circ }\left(\tau \right),\tau )d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ \hfill & ={\lambda }_0{\int }_0^T\ge 〈{\int }_T^t{\phi }_x(x^{\circ }\left(\tau \right),{\mu }^{\circ }\left(\tau \right),\tau )d\tau ,x^{\prime }\left(t\right)〉dt\hfill \\ \hfill & ={\lambda }_0{\int }_0^T{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)dt.\hfill \end{array}$$

(26)

The fourth term in (23) can be streamlined by applying the integration by parts formula for Stieltjes integrals (see [24]), together with the assumptions $g(z_0,0)<0$ and $g(x_1,T)<0$. In particular, since the boundary points are inactive ($0\notin R$ and $T\notin R$), it follows that $\mu \left(0\right)=\mu \left(T\right)=0$.

$${\int }_0^T〈x^{\prime }\left(t\right),{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)〉dt={\int }_0^T{g}_x(x^{\circ }\left(t\right),t)x\left(t\right)d\mu \left(t\right).$$

(27)

Then, by (24), (25), (26), and (27), we have that

$$\begin{array}{cc}\hfill -{\int }_0^T\langle \psi \left(t\right),x^{\prime }\left(t\right)\rangle dt& =-{\int }_0^T〈{\int }_0^t{K}_{xt}\left(t,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)x\left(\tau \right)d\tau ,\psi \left(t\right)〉dt\hfill \\ \hfill & -{\int }_0^T〈\psi \left(t\right),K_x(t,t,x^{\circ }\left(\tau \right),u^{\circ }\left(t\right)x\left(t\right)〉dt\hfill \\ \hfill & +{\lambda }_0{\int }_0^T{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)dt\hfill \\ \hfill & +{\int }_0^T{g}_x(x^{\circ }\left(t\right),t)x\left(t\right)d\mu \left(t\right)-\langle a,x\left(T\right)\rangle +{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)\hfill \end{array}$$

(28)

Then, rewriting last equality, we have

$$\begin{array}{cc}\hfill & {\lambda }_0{\int }_0^T{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)dt+{\int }_0^T{g}_x(x^{\circ }\left(t\right),t)x\left(t\right)d\mu \left(t\right)-\langle a,x\left(T\right)\rangle +{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)\hfill \\ \hfill & ={\int }_0^T〈{\int }_0^t{K}_{xt}\left(t,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)x\left(\tau \right)d\tau ,\psi \left(t\right)〉dt+{\int }_0^T〈\psi \left(t\right),K_x(t,t,x^{\circ }\left(\tau \right),u^{\circ }\left(t\right)x\left(t\right)〉dt\hfill \\ \hfill & -{\int }_0^T\langle \psi \left(t\right),x^{\prime }\left(t\right)\rangle dt\hfill \end{array}$$

Since $x\left(t\right)$ satisfies the variational Equation (19), we have that

$$\begin{array}{cc}\hfill -{\int }_0^T\langle \psi \left(t\right),x^{\prime }\left(t\right)\rangle dt& +{\int }_0^T〈{\int }_0^t{K}_{xt}\left(t,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)x\left(\tau \right)d\tau ,\psi \left(t\right)〉dt\hfill \\ \hfill & +{\int }_0^T〈\psi \left(t\right),K_x(t,t,x^{\circ }\left(\tau \right),u^{\circ }\left(t\right)x\left(t\right)〉dt\hfill \\ \hfill & ={\int }_0^T〈{\int }_0^t{K}_{ut}\left(t,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)u\left(\tau \right)d\tau ,\psi \left(t\right)〉dt\hfill \\ \hfill & +{\int }_0^T〈\psi \left(t\right),K_u(t,t,x^{\circ }\left(\tau \right),u^{\circ }\left(t\right)u\left(t\right)〉dt.\hfill \end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill & & {\lambda }_0{\int }_0^T{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)dt+{\int }_0^T{g}_x(x^{\circ }\left(t\right),t)x\left(t\right)d\mu \left(t\right)-\langle a,x\left(T\right)\rangle \hfill \\ & & +{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)\hfill \\ & & =-{\int }_0^T〈{\int }_0^t{K}_{ut}\left(t,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right)\right)u\left(\tau \right)d\tau ,\psi \left(t\right)〉dt\hfill \\ & & -{\int }_0^T〈\psi \left(t\right),K_u(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)u\left(t\right)〉dt\hfill \\ & & =-{\int }_0^T〈{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(s\right)ds,u\left(t\right)〉dt\hfill \\ & & -{\int }_0^T〈K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right),u\left(t\right)〉dt\hfill \\ & & =-{\int }_0^T〈{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(s\right)ds+K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right),u\left(t\right)〉dt\hfill \end{array}$$

(29)

Consequently, from the identity (22), we deduce

$$\begin{array}{cc}\hfill {\mathfrak{L}}_2^{\prime }\left(u\right)& ={\int }_0^T\langle -{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)\hfill \\ \hfill & +{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),u\left(t\right)\rangle dt,\hfill \end{array}$$

(30)

for every $(u\in L_{\infty }^r[0,T])$. Since ${\mathfrak{L}}_2^{\prime }$ acts as a supporting functional of ${\mho }_2^{\prime }$ at the point $u^{\circ }\in {\mho }_2^{\prime }$, Example 10.5 (see [21], p. 76) yields

$$〈-{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),U-u\left(t\right)〉\ge 0,$$

for all $U\in \Omega$ and for almost every $t\in [0,T]$.

We now argue that the alternative ${\lambda }_0=0$ together with $\psi =0$ cannot occur. Indeed, if $\psi =0$, then $\psi \left(T\right)=a=0$. Hence

$${\mathfrak{L}}_{12}(x,u)=\langle a,x\left(T\right)\rangle =0((x,u)\in E),$$

so ${\mathfrak{L}}_{12}\equiv 0$. From (12) with ${\lambda }_0=0$, it follows that

$${\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)=0,(t\in [0,T]),$$

which forces ${\mathfrak{L}}_3=0$. In addition, by (22), we also obtain ${\mathfrak{L}}_2^{\prime }\left(u\right)=0$ for $(v\in L_{\infty }^r[0,T])$; plugging this into (21) implies ${\mathfrak{L}}_{11}=0$, i.e.,

$${\mathfrak{L}}_1={\mathfrak{L}}_{11}+{\mathfrak{L}}_{12}=0,$$

contradicting Theorem 3.

Up to this point, two auxiliary hypotheses have been invoked:

First, we assumed ${\bigwedge }_0\ne \varnothing$. Second, we postulated that

$$x\left(t\right)={\int }_0^t{K}_x(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right))x\left(s\right)+K_u(t,s,x^{\circ }\left(s\right),u^{\circ }\left(s\right))u\left(s\right)ds,$$

is controllable.

We now show that these assumptions are dispensable. If ${\bigwedge }_0=\varnothing$, by the definition of ${\bigwedge }_0$, it follows that

$$\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\int }_0^T\left[{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(T\right)+{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)u\left(t\right)\right]dt=0.$$

Take $\mu =0$ and $a=0$. Then, from (29), we infer

$$\begin{array}{ccc}\hfill & & \nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\int }_0^T{\phi }_x(x^{\circ },u^{\circ },t)x\left(T\right)=\hfill \\ & -& {\int }_0^T〈{\int }_t^T{K}_{ut}^{*}(s,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(s\right)ds+K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right),u\left(t\right)〉dt,\hfill \end{array}$$

(31)

for all $(x,u)$ where x solves (19). Hence, for each $(u\in L_{\infty }^r[0,T])$,

$${\int }_0^T〈-{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),u\left(t\right)〉dt=0,$$

which implies

$$〈-{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)+{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),U-u\left(t\right)〉=0,$$

for all $U\in \Omega$ and almost every $t\in [0,T]$.

Finally, suppose (19) is not controllable. Then, by Lemma 1, there exists a nontrivial $\psi \in C^n[0,T]$ such that

$$\dot{\psi }\left(t\right)=-{\int }_t^T{K}_{xt}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_x^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right),$$

and, for all $t\in [0,T]$,

$${\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds+K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)=0.$$

Choosing ${\lambda }_0=0$ and $\mu =0$, the function $\psi$ satisfies (12), and therefore

$$〈-{\int }_t^T{K}_{ut}^{*}(s,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right)\psi \left(s\right)ds-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right),U-u^{\circ }\left(t\right)〉\ge 0,$$

for every $U\in \Omega$ and almost all $t\in [0,T]$.

The proof of Theorem 1 is now complete. □

5. Sufficient Conditions for Optimality

The first-order requirement in Theorem 1 (the maximum principle) becomes a sufficient criterion under additional hypotheses. To make this explicit, we focus on the specialization of Problem 1 in which the underlying Volterra dynamics are linear.

Problem 2.

$$h\left(x\left(T\right)\right)+{\int }_0^T\phi (x\left(t\right),u\left(t\right),t)dt⟶\mathrm{loc}min.$$

(32)

$$(x,u)\in E:=C^n[0,T]\times L_{\infty }^r[0,T],$$

$$x\left(t\right)={\int }_0^t\left[A(t,s)x\left(s\right)+B(t,s)u\left(s\right)\right]ds,$$

(33)

$$x\left(T\right)=x_1,x_1\in {\mathbb{R}}^n,$$

(34)

$$u\left(t\right)\in \Omega ,\forall t\in [0,T],$$

(35)

$$g\left(x\right(t),t)\le 0,t\in [0,T].$$

(36)

Here $x\left(t\right)\in {\mathbb{R}}^n$ is the state and $u\left(t\right)\in {\mathbb{R}}^r$ is the control. The kernels $A(t,s)$ and $B(t,s)$ have sizes $n\times n$ and $n\times r$, respectively, with entries of class $C^1(\Delta ;\mathbb{R})$, where

$$\Delta :=\{(t,s)\in {[0,T]}^2:0\le s\le t\le T\}.$$

In other words, A and B are continuously differentiable in both arguments on the triangular domain Δ. Let $(x^{\circ },u^{\circ })\in E$ be any feasible pair satisfying (33)–(36).

Theorem 7.

Assume that the hypotheses $\left(a\right)$–$\left(c\right)$ together with items $\left(\alpha \right)$ and $\left(\beta \right)$ of Theorem 1 are satisfied. In addition, suppose the following:

(A) 

The linear Volterra system (33) is controllable.

(B) 

There exists a control $\tilde{u}\in L_{\infty }^r[0,T]$ with $\tilde{u}\left(t\right)\in int(\Omega )$ for all $t\in [0,T]$.

(C) 

Let $\tilde{x}$ be the state associated with $\tilde{u}$ via (33). Then $\tilde{x}\left(T\right)=x_1$ and $g(\tilde{x}\left(t\right),t)<0$ for every $t\in [0,T]$.

(D) 

The functions h, g, and ψ are convex in their first two arguments.

Then, the pair $(x^{\circ },u^{\circ })$ is a global solution of Problem 2.

Proof. 

Define the mapping $\overline{\mathfrak{L}}:E\to \mathbb{R}$ by

$$\overline{\mathfrak{L}}(x,u)=h\left(x\left(T\right)\right)+{\int }_0^T\psi (x\left(t\right),u\left(t\right),t)dt.$$

Set $\mho :={\mho }_1\cap {\mho }_2\cap {\mho }_3$, where ${\mho }_1$ is given by (33) and (34), ${\mho }_2$ by (35), and ${\mho }_3$ by (36), as in Theorem 1.

With this notation, Problem 2 is equivalent to

$$\left\{\begin{array}{c}\overline{\mathfrak{L}}(x,u)⟶\mathrm{loc}min,\hfill \\ \\ (x,u)\in \mho .\hfill \end{array}\right.$$

Observe that each ${\mho }_i$ ($i=1,2,3$) is convex; moreover, by assumptions (C)–(D) the functional $\overline{\mathfrak{L}}$ is convex and $(\tilde{x},\tilde{u})\in int\left({\mho }_2\right)\cap int\left({\mho }_3\right)\cap {\mho }_1$.

Therefore, by Theorem 2.17 in [14], we have the following:

• $(x^{\circ },u^{\circ })$ minimizes $\overline{\mathfrak{L}}$ over ℧ if and only if there exist ${\mathfrak{L}}_i\in {\bigwedge }_i^{+}$ ($i=0,1,2,3$), not all vanishing, such that

$${\mathfrak{L}}_0+{\mathfrak{L}}_1+{\mathfrak{L}}_2+{\mathfrak{L}}_3=0.$$

Here ${\bigwedge }_i$ ($i=0,1,2,3$) are the approximation cones defined exactly as in Theorem 1.

Let ${\bigwedge }_3={\bigwedge }_a({\mho }_3,(x^{\circ },u^{\circ }))$ denote the feasible direction cone to ${\mho }_3$ at $(x^{\circ },u^{\circ })$. Then

$${\bigwedge }_3\supset {\bigwedge }_d\left(\overline{\mathfrak{L}},(x^{\circ },u^{\circ })\right)=:{\bigwedge }_d,$$

where

$${\bigwedge }_d:=\left\{(x,u)\in E:\langle g_x(x^{\circ }\left(t\right),t,\alpha ),x\left(t\right)\rangle <0(t\in R)\right\},$$

and

$$R:=\{t\in [0,T]:g(x^{\circ }\left(t\right),t)=0\}.$$

Hence, by dual-cone monotonicity,

$${\bigwedge }_3^{+}\subset {\bigwedge }_d^{+}.$$

Consequently, any $\mathfrak{L}\in {\bigwedge }_d^{+}$ has the representation

$$\mathfrak{L}(x,u)=-{\int }_0^T\langle g_x(x^{\circ }\left(t\right),t),x\left(t\right)\rangle d\mu \left(t\right),((x,u)\in E),$$

for some non-negative Borel measure $\mu$ supported on R.

Now, assume the maximum principle of Theorem 1 holds. Then, there exist ${\lambda }_0\ge 0$, $a\in {\mathbb{R}}^n$, and a non-negative Borel measure $\mu$ supported on R, together with a function $\psi \in L_1^n[0,T]$ solving

$$\begin{array}{ccc}\hfill -\psi \left(t\right)& =& -{\int }_t^T\left\{{\int }_{\tau }^T{\displaystyle \frac{\partial A^{*}}{\partial t}}(s,t)\psi \left(s\right)ds-{\lambda }_0{\phi }_x(x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right),\tau )\right\}d\tau \hfill \\ \hfill & -& {\int }_t^T{A}^{*}(\tau ,\tau )\psi \left(\tau \right)d\tau +{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)-a+{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right),\hfill \end{array}$$

(37)

with $({\lambda }_0,\psi )\ne (0,0)$, and for every $U\in \Omega$ and almost every $t\in [0,T]$,

$$〈-{\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds-B^{*}(t,t)\psi \left(t\right)+{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),U-u\left(t\right)〉\ge 0.$$

(38)

To conclude, it suffices to exhibit ${\mathfrak{L}}_i\in {\bigwedge }_i^{+}$ ($i=0,1,2,3$), not all zero, such that ${\mathfrak{L}}_0+{\mathfrak{L}}_1+{\mathfrak{L}}_2+{\mathfrak{L}}_3=0$. Define

$${\mho }_2^{\prime }=\{u\in L_{\infty }^r:u\left(t\right)\in \Omega \mathrm{for}\mathrm{all}t\in [0,T],\mathrm{a}.\mathrm{e}.\},$$

and the functionals

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_2^{\prime }& :& L_{\infty }^r\to \mathbb{R},\hfill \\ \hfill {\mathfrak{L}}_2^{\prime }\left(u\right)& =& {\int }_0^T〈-{\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds-B^{*}(t,t)\psi \left(t\right)+{\lambda }_0{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t),u\left(t\right)〉dt,\hfill \\ \hfill {\mathfrak{L}}_2& :=& (0,{\mathfrak{L}}_2^{\prime }).\hfill \end{array}$$

From (38), we obtain

$${\mathfrak{L}}_2^{\prime }\left(u\right)\ge {\mathfrak{L}}_2^{\prime }\left(u^{\circ }\right)(u\in {\mho }_2^{\prime }),$$

so ${\mathfrak{L}}_2^{\prime }$ supports ${\mho }_2^{\prime }$ at $u^{\circ }$, and therefore ${\mathfrak{L}}_2=(0,{\mathfrak{L}}_2^{\prime })\in {\bigwedge }_2^{+}$. Define ${\mathfrak{L}}_{11}:E\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_{11}(x,u)& =& {\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\lambda }_0{\int }_0^T[{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)\hfill \\ & +& {\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)u\left(t\right)]d-{\mathfrak{L}}_2^{\prime }\left(u\right)-\langle a,x\left(T\right)\rangle \hfill \\ & +& {\int }_0^T\langle g_x(x^{\circ }\left(t\right),t),x\left(t\right)\rangle d\mu \left(t\right).\hfill \end{array}$$

We claim ${\mathfrak{L}}_{11}\in L_1^{+}$, where

$$L_1=\left\{(x,u):x\left(t\right)={\int }_0^t\left[A(t,\tau )x\left(\tau \right)+B(t,\tau )u\left(\tau \right)\right]d\tau ,t\in [0,T]\right\},$$

as in Theorem 1. Indeed, for $(x,u)\in L_1$, multiply (37) by $\dot{x}\left(t\right)$ and integrate over $[0,T]$ to obtain

$$\begin{array}{cc}\hfill & {\lambda }_0{\int }_0^T{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)dt+{\int }_0^T\langle g_x(x^{\circ }\left(t\right),t),x\left(t\right)\rangle d\mu \left(t\right)-\langle a,x\left(T\right)\rangle \hfill \\ \hfill & +{\lambda }_0\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)=-{\int }_0^T〈{\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds+B^{*}(t,t)\psi \left(t\right),u\left(t\right)〉dt.\hfill \end{array}$$

Hence

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_{11}(x,u)& =& -{\mathfrak{L}}_2^{\prime }\left(u\right)-{\int }_0^T〈{\int }_t^T{\displaystyle \frac{\partial B^{*}}{\partial t}}(s,t)\psi \left(s\right)ds+B^{*}(t,t)\psi \left(t\right),u\left(t\right)〉dt\hfill \\ & & +{\lambda }_0{\int }_0^T{\psi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)u\left(t\right)dt,\hfill \end{array}$$

so that

$${\mathfrak{L}}_{11}(x,u)=-{\mathfrak{L}}_2^{\prime }\left(u\right)+{\mathfrak{L}}_2^{\prime }\left(u\right)=0,$$

and therefore ${\mathfrak{L}}_{11}\in L_1^{+}$.

Finally, define

$${\mathfrak{L}}_0,{\mathfrak{L}}_1,{\mathfrak{L}}_3:E\to \mathbb{R},$$

by

$$\begin{array}{ccc}& & {\mathfrak{L}}_0(x,u)=\hfill \\ & & {\lambda }_0\left\{\nabla h\left(x^{\circ }\left(T\right)\right)x\left(T\right)+{\int }_0^T\left[{\phi }_x(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)x\left(t\right)+{\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)u\left(t\right)\right]dt\right\},\hfill \\ & & {\mathfrak{L}}_1(x,u)={\mathfrak{L}}_{11}(x,u)+\langle a,x\left(T\right)\rangle ,\hfill \\ & & {\mathfrak{L}}_3(x,u)={\int }_0^T\langle g_x(x^{\circ }\left(t\right),t),x\left(t\right)\rangle d\mu \left(t\right).\hfill \end{array}$$

Then ${\mathfrak{L}}_0\in {\bigwedge }_0^{+}$, ${\mathfrak{L}}_1\in {\bigwedge }_1^{+}$, ${\mathfrak{L}}_3\in {\bigwedge }_3^{+}$, and

$${\mathfrak{L}}_0+{\mathfrak{L}}_1+{\mathfrak{L}}_2+{\mathfrak{L}}_3=0,$$

with at least one of these functionals being nonzero because, by hypothesis, $({\lambda }_0,\psi )\ne (0,0)$.

The claimed global minimality of $(x^{\circ },u^{\circ })$ follows from the convexity assumptions. □

6. A Mathematical Model

In this part, we outline a representative real-world model where our results apply; the subsequent section states an open question.

Optimal Control for an Epidemic: The SIR Scheme

Consider a community facing an infectious disease that we aim to mitigate by vaccination. We introduce the state components:

• $I\left(t\right)$: Number of infectious individuals capable of transmitting the disease;
• $S\left(t\right)$: Number of susceptible (yet noninfected) individuals;
• $R\left(t\right)$: Number of removed/recovered individuals who are no longer susceptible.

Let $r>0$ denote the transmission coefficient, $\gamma >0$ the recovery rate, and $u\left(t\right)$ the vaccination/control input, subject to the bound $0\le u\left(t\right)\le e$. The associated optimal control problem for the SIR dynamics is

$${\displaystyle \frac{\alpha }{2}}I\left(T\right)+{\int }_0^T{\displaystyle \frac{1}{2}}u{\left(t\right)}^{2}dt⟶\mathrm{loc}min,$$

(39)

$$\left\{\begin{array}{ccc}S\left(t\right)\hfill & =\hfill & {\displaystyle S_0\left(t\right)+{\int }_0^t\left(-rS\left(\tau \right)I\left(\tau \right)+u\left(\tau \right)\right)d\tau ,}\hfill \\ I\left(t\right)\hfill & =\hfill & {\displaystyle I_0\left(t\right)+{\int }_0^t\left(rS\left(\tau \right)I\left(\tau \right)-\gamma I\left(\tau \right)-u\left(\tau \right)\right)d\tau ,}\hfill \\ R\left(t\right)\hfill & =\hfill & {\displaystyle R_0\left(t\right)+{\int }_0^t\gamma I\left(\tau \right)d\tau ,}\hfill \end{array}\right.$$

(40)

$$g\left(x\left(t\right),t\right)\le 0,t\in [0,T],$$

(41)

$$u\left(t\right)\in [0,e],t\in [0,T]\mathrm{a}.\mathrm{e}.,$$

(42)

where $\alpha >0$.

In contrast with classical formulations using differential equations, we model the epidemic process through an integral representation. This approach naturally accounts for the cumulative nature of epidemic transitions and facilitates numerical implementation.

The functions $S_0\left(t\right)$, $I_0\left(t\right)$, and $R_0\left(t\right)$ should not be interpreted merely as constant initial values. Instead, they may represent either

• Pre-processed or interpolated empirical data on the susceptible, infected, and recovered subpopulations prior to the onset of control;
• Accumulated trajectories obtained from previous simulation stages or observational time series;
• Non-constant baseline trajectories reflecting uncertainty or memory effects.

In epidemic models such as the SIR system, various forms of time-dependent state constraints of the form

$$g\left(S\right(t),I(t),R(t),t)\le 0,$$

may arise to reflect practical limitations, policy objectives, or ethical considerations. Below, we describe several representative types of such constraints.

• Healthcare Capacity Constraints:

  –

  Limit on active infections:

  $$g(S,I,R,t)=I\left(t\right)-I_{max}\le 0.$$

  –

  Maximum proportion infected:

  $$g(S,I,R,t)={\displaystyle \frac{I\left(t\right)}{S\left(t\right)+I\left(t\right)+R\left(t\right)}}-\alpha \le 0.$$

• Containment or Safety Constraints:

  –

  Infected individuals less than susceptible:

  $$g(S,I,R,t)=I\left(t\right)-S\left(t\right)\le 0.$$

  –

  Infection rate constraint:

  $$g(S,I,R,t)=I\left(t\right)-\delta \le 0.$$

• Final-Time Constraints:

  –

  Minimum number of recovered individuals at final time:

  $$g(S,I,R,T)=R_{\mathrm{target}}-R\left(T\right)\le 0.$$

  –

  Near eradication at the final time:

  $$g(S,I,R,T)=I\left(T\right)-\epsilon \le 0.$$

• Mixed-Variable Constraints:

  –

  Combined restriction on susceptible and recovered populations:

  $$g(S,I,R,t)=S\left(t\right)-R\left(t\right)-\beta I\left(t\right)\le 0$$

  –

  Preservation of a minimum level of susceptibles:

  $$g(S,I,R,t)=\sigma -S\left(t\right)\le 0.$$

• Cost- or Resource-Based Constraints:

  –

  Indirect economic/social cost limitation:

  $$g(S,I,R,t)=c_{1}S\left(t\right)+c_{2}I\left(t\right)+c_{3}R\left(t\right)-B\le 0.$$

This formulation allows greater flexibility in modeling and aligns well with the structure of optimal control theory in Banach spaces, where integral equations provide a natural framework for weak formulations and the application of variational methods.

The aim is to select a vaccination policy that minimizes the functional objective over the fixed horizon T. With the notation introduced below, this problem can be put into the abstract framework of Theorem 1.

$$\begin{array}{cc}\hfill x& =\left(\begin{array}{c}S\\ I\\ R\end{array}\right),K(t,s,x,U)=\left(\begin{array}{c}-rSI+U\\ rSI-\gamma I-U\\ \gamma I\end{array}\right),h\left(x\right)={\displaystyle \frac{1}{2}}\alpha Iand\phi (x,u,t)={\displaystyle \frac{1}{2}}{U}^2.\hfill \end{array}$$

$$\begin{array}{cc}\hfill x\left(t\right)& =p\left(t\right)+{\int }_0^{t}K(t,s,x\left(s\right),u\left(s\right))ds\hfill \\ \hfill & =p\left(t\right)+{\int }_0^t\left(\begin{array}{c}-rS\left(s\right)I\left(s\right)+u\left(s\right)\\ rS\left(s\right)I\left(s\right)-\gamma I\left(s\right)-u\left(s\right)\\ \gamma I\left(s\right)\end{array}\right)ds,\hfill \\ \hfill p\left(t\right)& =\left(\begin{array}{c}{S}_0\left(t\right)\\ I_0\left(t\right)\\ R_0\left(t\right)\end{array}\right)\hfill \end{array}$$

where $(x,u)\in E=\mathcal{C}([0,T];{IR}^3)\times L_{\infty }([0,T];IR)$.

Therefore, the adjoint equation becomes the following equation:

$$\begin{array}{ccc}\hfill -\psi \left(t\right)& =& {\int }_t^T-K_x^{*}(\tau ,\tau ,x^{\circ }\left(\tau \right),u^{\circ }\left(\tau \right))\psi \left(\tau \right)d\tau +\nabla h\left(\left(x^{\circ }\left(T\right)\right)\right)+{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)\hfill \\ \\ & =& {\int }_t^T\left(\begin{array}{ccc}r{I}^{\circ }\left(\tau \right)& -r{I}^{\circ }\left(\tau \right)& 0\\ r{S}^{\circ }\left(\tau \right)& -r{S}^{\circ }\left(t\tau \right)+\gamma & 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{\psi }_1\left(\tau \right)\\ {\psi }_2\left(\tau \right)\\ {\psi }_3\left(\tau \right)\end{array}\right)d\tau +\left(\begin{array}{c}0\\ {\displaystyle \frac{\alpha }{2}}\\ 0\end{array}\right)+{\int }_t^T{g}_x(x^{\circ }\left(\tau \right),\tau )d\mu \left(\tau \right)\hfill \end{array}$$

Since no terminal constraint is imposed on $x\left(T\right)$, we may set $a=0$. For convenience, we also normalize the multiplier to ${\lambda }_0=1$. We now state the Pontryagin maximum condition.

From

$$-K_u^{*}(t,t,x^{\circ }\left(t\right),u^{\circ }\left(t\right))\psi \left(t\right)=\left(\begin{array}{ccc}-1& 1& 0\end{array}\right)\left(\begin{array}{c}{\psi }_1\left(t\right)\\ {\psi }_2\left(t\right)\\ {\psi }_3\left(t\right)\end{array}\right)={\psi }_2\left(t\right)-{\psi }_1\left(t\right),$$

and since ${\phi }_u(x^{\circ }\left(t\right),u^{\circ }\left(t\right),t)=u^{\circ }\left(t\right)$, the variational inequality reads

$$〈{\psi }_2\left(t\right)-{\psi }_1\left(t\right)+u^{\circ }\left(t\right),U-u^{\circ }\left(t\right)〉\ge 0,\forall U\in \Omega ,\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,T].$$

Then

$$\underset{U\in [0,e]}{max}({\psi }_1\left(t\right)-{\psi }_2\left(t\right)-u^{\circ }\left(t\right))U=({\psi }_1\left(t\right)-{\psi }_2\left(t\right)-u^{\circ }\left(t\right))u^{\circ }\left(t\right).$$

Hence, optimal control is the projection of ${\psi }_1\left(t\right)-{\psi }_2\left(t\right)$ onto $[0,e]$; i.e.,

$$u^{\circ }\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{\psi }_1\left(t\right)-{\psi }_2\left(t\right)<0,\hfill \\ {\psi }_1\left(t\right)-{\psi }_2\left(t\right),\hfill & \mathrm{if}0\le {\psi }_1\left(t\right)-{\psi }_2\left(t\right)\le e,\hfill \\ e,\hfill & \mathrm{if}{\psi }_1\left(t\right)-{\psi }_2\left(t\right)>e.\hfill \end{array}\right.$$

At the terminal time, we have ${\psi }_1\left(T\right)-{\psi }_2\left(T\right)=\alpha /2$. Therefore, if $2e<\alpha$, the control saturates at the upper bound near T, i.e., $u^{\circ }\left(t\right)=e$ in a neighborhood of the final time.

7. Open Problem

We close with an open question that outlines a fertile line of inquiry, one that could naturally evolve into a doctoral dissertation. The challenge concerns an optimal control setting that, at the same time, accommodates impulsive interventions and enforces pathwise constraints on the state. Our aim is to investigate the following formulation in forthcoming work.

Problem 3.

Determine a control $u\in L^{\infty }([0,T];{\mathbb{R}}^r)$ and a trajectory $x\in \mathcal{PW}([0,T];{\mathbb{R}}^n)$ that locally minimize the performance index

$$h\left(x\left(T\right)\right)+{\int }_0^T\phi (x\left(t\right),u\left(t\right),t)dt⟶\underset{\mathrm{loc}}{min}.$$

(43)

This is subject to the following requirements.

• Volterra-type state evolution:

  $$x\left(t\right)=p\left(t\right)+{\int }_0^{t}K\left(t,s,x\left(s\right),u\left(s\right)\right)ds,$$

  (44)

  where the kernel K satisfies suitable regularity and growth hypotheses (to be specified in the sequel).
• Terminal equalities:

  $$G_i\left(x\left(T\right)\right)=0,i=1,2,\dots ,q\le n.$$

  (45)
• Impulsive dynamics:

  $$x\left(t_k^{+}\right)=x\left(t_k^{-}\right)+{\mathcal{J}}_k\left(x\left(t_k\right)\right),k=1,2,\dots ,p,$$

  (46)

  with prescribed impulse times $\left\{t_k\right\}\subset [0,T]$ and given jump maps ${\mathcal{J}}_k$.
• Admissible controls:

  $$u\left(t\right)\in \Omega \subset {\mathbb{R}}^r\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,T],$$

  (47)

  where Ω is compact and convex.
• Pathwise state restriction:

  $$g\left(x\left(t\right),t\right)\le 0,\forall t\in [0,T].$$

  (48)

8. Conclusions and Final Remarks

We have developed a fresh criterion for the controllability of linear systems driven by Volterra integral equations, obtained via a new lemma that is of standalone mathematical interest. A direct consequence is that the customary hypothesis requiring controllability of the linearized (variational) dynamics near an optimal pair turns out to be unnecessary.

On this foundation, we extended Pontryagin’s maximum principle to a wide class of optimal control problems with Volterra-type dynamics, allowing for both terminal constraints and time-dependent path constraints. Leveraging the Dubovitskii–Milyutin framework, we established necessary optimality conditions under mild regularity. In addition, Section 5 (Theorem 7) provides sufficient conditions for optimality: under convexity of the cost functional and linear Volterra dynamics, the maximum principle becomes a sufficient criterion for global optimality, recovering the classical sufficiency in the differential case (Corollary 1). Two complementary adjoint representations were derived: (i) a Volterra adjoint equation when only terminal constraints are present and (ii) an adjoint relation involving a Stieltjes integral against a non-negative Borel measure in the presence of state constraints.

When the Volterra kernel collapses to a differential operator, our statements specialize to the classical Pontryagin maximum principle, thereby unifying the integral and differential settings within a single theory. The case study on optimal vaccination in an SIR epidemic model highlights the practical reach of the results.

We also proposed an open problem that naturally follows from this work, pointing to fertile directions for future research and potential thesis-level investigations. Progress on these questions could further broaden the scope of the Dubovitskii–Milyutin approach and stimulate additional applications.