NECESSARY CONDITIONS FOR A WEAK MINIMUM IN A GENERAL OPTIMAL CONTROL PROBLEM WITH INTEGRAL EQUATIONS ON A VARIABLE TIME INTERVAL

. We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonﬁxed time interval, subject to end- point constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain ﬁrst-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary diﬀerential equations.

1. Introduction. It is commonly known that, for problems with ordinary differential equations (ODEs), the theory of first order necessary optimality conditions including the Pontryagin maximum principle is now completely developed. It covers problems both on a fixed and a nonfixed time intervals containing pure state and mixed state-control constraints, as well as different types of integral and endpoint constraints. A challenging question arises about similar complete theory for problems with control system given by Volterra-type integral equations. Such equations could be considered as a close generalization of ODEs, since, in some respects, they possess similar properties. However, this similarity is not complete, and the integral equations have also some essential specificity that differ them from the ODEs. Nevertheless, like the calculus of variations and optimal control for ODEs, the corresponding theory for Volterra-type equations should desirably include necessary conditions both for the strong and weak local minima.

ANDREI V. DMITRUK AND NIKOLAI P. OSMOLOVSKII
First results on optimality conditions for problems with Volterra-type equations appeared soon after publication of the famous monograph by Pontryagin and his collaborators [16], and concerned the maximum principle. These results were due to Vinokurov [17], Bakke [1], Neustadt [15], Kamien and Muller [12], Hartl and Sethi [11], and Carlson [4]. All of them considered problems on a fixed time interval, without mixed or state constraints. Later, De la Vega [5] considered a time optimal control problem, which was the first publication concerning problems with integral equations on a nonfixed time interval. After that the interest to problems with integral equations arose again, and new results, including second-order conditions for state constrained problems, were obtained by Bonnans, Dupuis, and De la Vega [3]. In [8], we considered a general problem with state and mixed constraints on a fixed time interval and obtained necessary conditions for the weak minimum. If the time interval is nonfixed, the notion of weak minimum should be slightly generalized to the notion of extended weak minimum, which takes into account also the variations of time. Necessary conditions for this type of minimum, in problems without state or mixed constraints were obtained in our recent paper [9].
In the present paper we consider a general problem combining both a nonfixed time interval and state and mixed constraints, and obtain first order conditions for an extended weak minimum. They are given in Theorem 3.1. As far as we know, such conditions for problems with integral equations on a variable time interval subject to state and mixed constraints were not obtained up to now. Note that they are not a direct combination of the results in [8,9]. Based on those former results it is hardly possible to give even the proper formulation of the present results. Their novelty, as compared with those for problems on a fixed time interval is that the costate equation and the terminal transversality condition with respect to t involve nonstandard terms that are absent in problems with ODEs. Along with this, the presence of state and mixed constraints in problems on a nonfixed time interval (unlike in problems on a fixed interval) leads to additional delicacy not only in the proofs, but even in the formulations.
As was already mentioned in [8], necessary conditions for the weak (or extended weak) minimum in optimal control problems constitute an important stage in derivation of any further necessary optimality condition, including maximum principle or higher order conditions, and thus, they deserve a separate thorough study for each specific class of problems, like it is done in the classical calculus of variations. This is why we focus on these conditions. Following the tradition, we call them stationarity conditions (or local maximum principle ).
The paper is organized as follows. In Section 2 we formulate a general optimal control problem with integral equations on a variable time interval, called Problem A. We also define in this section the notion of extended weak minimum. Section 3 is devoted to formulation of the main result of the paper -the first order necessary condition for an extended weak minimum in Problem A (Theorem 1).
The proof of Theorem 1 is based on a reduction of Problem A to a problem of new type B on a fixed time interval, stated in Section 4. The control system in Problem B has a more general type than that in Problem A. In Theorem 2 of this section we present the Lagrange multipliers rule (LMR) for a class of abstract nonsmooth optimization problems which contains Problem B as a special case. In Section 5, we apply LMR to Problem B, and then, in Section 6, performing some analysis of LMR, obtain the local maximum principle in Problem B. This important intermediate result is given in Theorem 3. Finally, in Section 7, we finish the proof where x(·) is a continuous n− dimensional and u(·) a measurable essentially bounded r− dimensional vector-function on [t 0 , t 1 ]. As usual, we call x the state variable and u the control variable (or simply the control ). We assume for simplicity that the function f is defined and twice continuously differentiable on an open set R ⊂ R 2+n+r .
The problem is to minimize the endpoint cost functional on the set of solutions of system (1) satisfying the endpoint constraints the mixed state-control constraints and the pure state constraints where the functions ϕ 0 , ϕ i , η j are assumed to be defined and continuously differentiable on an open set P ⊂ R 2n+2 , and the functions F i , G j , Φ k are defined and continuously differentiable on an open set Q ⊂ R 1+n+r . (The Φ k can be formally considered as functions of three variables t, x, u.) The notation d(ϕ), d(η), d(F ), etc. stand for the numbers of these functions. Moreover, we impose the following Assumption RMC (on the regularity of mixed constraints). The mixed constraints (5)- (6) are regular in the following sense: at any point (t, x, u) ∈ Q satisfying relations F i ≤ 0 ∀ i and G j = 0 ∀ j, the system of vectors x, u) = 0 } is the set of active indices of mixed inequality constraints at the given point. Recall that a system consisting of two tuples of vectors p 1 , . . . , p m and q 1 , . . . q k in the space R r is said to be positively-linearly independent if there does not exist a nontrivial tuple of multipliers α 1 , . . . , α m , β 1 , . . . β k with all α i ≥ 0 such that The problem (1)-(7) will be called Problem A, and the relations (2)-(4) its endpoint block.
where the last integral shows, in a sense, how "far" we are from an ordinary differential equation. (Here f t means the partial derivative of the function f (t, s, x, u) with respect to the first, outer time variable t. ) If f does not depend on the outer time t, i.e., f = f (s, x(s), u(s)), then this integral disappears, and Problem A becomes a standard optimal control problem with the ODEẋ(t) = f (t, x(t), u(t)).
Obviously, each pair (x(t), u(t)) under consideration must "lie" in the domain R of the function f (t, s, x, u), i.e.
We will need even a stronger condition.
Definition 2.1. A pair of functions w(t) = (x(t), u(t)) defined on an interval t ∈ [t 0 , t 1 ] (with continuous x(t) and measurable essentially bounded u(t) ) will be called a process in Problem A if it satisfies (1) and its "extended graph" lies in the set R with some "margin", i.e., dist ((t, s, x(s), u(s)), ∂R) ≥ const > 0 for a.a. (t, s) ∈ ∆[t 0 , t 1 ], (9) or equivalently, there exists a compact set Ω ⊂ R such that (t, s, x(s), u(s)) ∈ Ω for a.a. (t, s) ∈ ∆[t 0 , t 1 ]. A process in problem A is called admissible if it satisfies all the constraints of the problem.
Like in any problem on a nonfixed time interval, the notion of weak minimum in Problem A needs a modification. Definition 2.2. We will say that an admissible process provides the extended weak minimum if there exists an ε > 0 such that, for any Lipschitz continuous bijective mapping ρ : [t 0 ,t 1 ] → [t 0 , t 1 ] satisfying the conditions |ρ(t) − t| < ε and |ρ(t) − 1| < ε, and for any admissible process w(t) = (x(t), u(t)), t ∈ [t 0 , t 1 ], satisfying the conditions the following inequality holds: J(w) ≥ J(w 0 ). (Notation (∀ ) conveniently means "for almost all".) The conditions on ρ imply ρ(t 0 ) =t 0 and ρ(t 1 ) =t 1 with |t 0 − t 0 | < ε and |t 1 −t 1 | < ε. If the interval [t 0 , t 1 ] is fixed and we take ρ(t) = t, then relations (11) describe the usual uniform closeness between the processes w 0 and w both in the state and control variables. However, since ρ(t) is variable, relations (11) extend the set of "competing" processes by allowing perturbations of the time, and thus, even for a fixed time interval, the extended weak minimum is, in general, stronger than the usual weak minimum.
3. Local maximum principle in Problem A. Let a process (10) provide the extended weak minimum in Problem A. We assume that the endpoints of the reference state x 0 (t) do not lie on the boundary of state constraints. To be more precise, that To formulate optimality conditions, let us introduce a tuple of Lagrange multipliers corresponding to all the constraints and the cost of Problem A: where α = (α 0 , α 1 , . . . , α d(ϕ) ) ∈ R d(ϕ)+1 with α i ≥ 0 ∀ i (for short, we will simply write α ≥ 0), and β = (β 1 , . . . , β d(η) ) ∈ R d(η) are vectors, are measurable bounded functions. We denote by dψ x , dψ t , dµ k the Lebesgue-Stieltjes measures which correspond to the functions ψ x , ψ t , µ k , respectively. These measures have no atoms at the pointst 0 andt 1 , and obviously, dµ k ≥ 0, k = 1, . . . , d(Φ). Byμ k (t) we denote the generalized derivative with respect to t of monotone nondecreasing function µ k (t), henceμ k (t) dt = dµ k (t). Similarly, ψ x (t) andψ t (t) denote generalized derivatives of functions ψ x (t) and ψ t (t), respectively. In what follows, all pointwise relations involving continuous functions hold for any t, and those involving measurable functions hold for almost all t. Further, introduce the modified Pontryagin function (here, ψ x f is the product of the row and column n− vectors), and the augmented (or extended) modified Pontryagin function Also, introduce the endpoint Lagrange function Both these functions refer to the tuple (13). In view of equation (8), define the function The functions H, H, l, and R will be used in formulation of optimality conditions. In what follows, we will need the expression of partial derivative of the functions H(t, s, x, u) with respect to the second, inner variable s along the process x 0 (t), u 0 (t) : (here f s is the partial derivative of the function f (t, s, x, u) with respect to the second, inner variable s, and f ts is its second mixed partial derivative), and therefore because, in order to take the derivative of F i (s, x, u) w.r.t. s, one should take its derivative w.r.t. its first argument, i.e. F it , and similar for G j (s, x, u) and Φ k (s, x, u) . For the process (10) and tuple (13) with the specified properties, let us formulate the conditions of local maximum principle (or the stationarity conditions): a) the nonnegativity conditions b) the nontrivality condition c) the endpoint complementary slackness conditions d) the pointwise complementary slackness conditions or equivalently, g) the transversality conditions in x, h) the transversality conditions in t, i) the stationarity condition of the extended Pontryagin function with respect to the control k) and the "energy evolution law" We call it in this way, since together with (24) it gives the equation for evolution of the function H(ψ x (t), t, x 0 (t), u 0 (t)) = −ψ t (t), which is often (especially in mechanical problems) regarded as the total energy of the system: (If the state and mixed constraints are absent and the dynamics does not explicitly depend on time: f = f (x, u), then H = H, R = 0, and we get the convenient "energy conservation law": H = const along the optimal process.) Note that using the generalized derivatives of functions of bounded variation, one can represent the adjoint equation in x and t in an easy-to-remember form: The main result of the paper is the following provides the extended weak minimum in Problem A and satisfies assumption (12), then there exists a tuple of multipliers (α, β, ψ x , ψ t , h i , m j , µ k ) satisfying the specified above properties and such that conditions a)-k) of the local maximum principle hold true.
Like in our previous paper [9], in order to prove Theorem 3.1, we reduce Problem A to an auxiliary problem on a fixed time interval by the change of time variable t = t(τ ), where dt/dτ = v(τ ) and v(τ ) > 0. The obtained auxiliary problem is a problem of type B described in the next section. We will derive optimality conditions in Problem B, apply them to the auxiliary problem, and rewrite the obtained conditions in terms of the original Problem A. Let us pass to the realization of this plan. 4. Problem B on a fixed time interval. Consider a system of the following form on a fixed interval [t 0 , t 1 ] : where x(t) and y(t) are continuous functions of dimensions n and m respectively, u(t) is a measurable and essentially bounded function on [t 0 , t 1 ]. Here we still denote the time by t. Like before, the data functions g and h are assumed to be twice continuously differentiable on an open set R ⊂ R 2+2m+n+r (if h is formally considered as a function of six variables t, s, y(t), y(s), x, u). Note, that this system does not fall into the framework of equation (1), since the integrand of the first equation depends on y(t), which can be regarded as the outer state variable, while y(s) is the inner state. Thus, we have to study a new, broader than (1), class of integral control systems.
Adding to the obtained system the mixed constraints, the state constraints and the terminal block, we obtain the following Problem B on a fixed interval [t 0 , t 1 ] : on the set of solutions w = (y, x, u) to system (31)-(32) satisfying the constraints one has to minimize the endpoint cost functional Like before, the functions η j , ϕ i , ϕ 0 are continuously differentiable on an open set P ⊂ R 2n+2m , and the functions F i , G j , Φ k continuously differentiable on an open set Q ⊂ R 1+m+n+r (considering Φ k as functions of four variables t, y, x, u). Moreover, we assume that the mixed constraints (33) and (34) are regular in the same sense as in Problem A. By definition, an admissible process in Problem B is any triple (y(t), x(t), u(t)), t ∈ [t 0 , t 1 ], satisfying all the constraints of this problem, whose extended graph lies in R with some margin.
We consider Problem B as a particular case of an abstract nonsmooth problem in a Banach space, hence we can apply the well known abstract Lagrange multipliers rule for nonsmooth problems. For the reader's convenience, we recall its formulation.
Lagrange multipliers rule for an abstract nonsmooth problem. Let X, Y, . . , ν, and g : D → Y given mappings. Consider the following problem We study the question of a local minimum at an admissible point x 0 ∈ D. Assume that the cost f 0 and the mappings b i are Frechet differentiable at x 0 , the operator g is strictly differentiable at x 0 , and the image of g (x 0 ) is closed (the weak regularity of equality constraint). Let K * i be the dual cone and K 0 i = −K * i the polar cone to K i , i = 1, . . . , ν. The following theorem holds [8].
. . , ν, and y * ∈ Y * , not all equal to zero, satisfying the complementary slackness conditions and such that the Lagrange function is stationary at x 0 : L (x 0 ) = 0.

5.
Lagrange multipliers rule for problem B. Let us apply Theorem 4.1 to Problem B which we represent in the form (39). In this problem, the role of X is played by the space with elements w = (y, x, u) and the norm ||w|| = ||y|| C + ||x|| C + ||u|| ∞ . The local minimum in this norm is exactly the weak minimum. The corresponding open set D ⊂ W is defined by the open sets P and R.
In what follows, w 0 = (y 0 , x 0 , u 0 ) ∈ W is a point of weak minimum in Problem B. Note that because of (31)-(32), the functions x 0 and y 0 are Lipschitz continuous.
The smoothness assumptions on the data functions are obviously fulfilled in D. Indeed, the cost functional (38) and the endpoints inequalities (37) are smooth, the mixed inequalities (33) have the form where Ω is the cone C − ([t 0 , t 1 ], R) of nonpositive functions in the Banach space C([t 0 , t 1 ], R) (one and the same for all k ). The mappings a i (w) and b k (w) are obviously smooth.
Weak regularity of the equality constraints. So, we only have to check the weak regularity assumption for the equality constraints, i.e. that the derivative of corresponding operator has a closed image. We will need the following property. Let an n × n− matrix Q(t, s) defined for (t, s) ∈ [t 0 , t 1 ] × [t 0 , t 1 ] be measurable and bounded in s, and continuous in t uniformly with respect to s. Consider the linear integral operator P : The operator P is surjective.
This fact is well-known, see e.g. [8]. We will use it below. Define the spaces The operator of equality constraints consists of four components: g(t, s, y(t), y(s), x(s), u(s)) ds = z 1 (t), h(t, s, y(s), u(s)) ds = z 2 (t), The equality constraints of Problem B can be represented as T (w) = 0. The derivative of this operator at the point w 0 has the form: [g yt (t, s)y(t) + g ys (t, s)y(s) + g x (t, s)x(s) + g u (t, s)u(s)] ds = z 1 (t), (40) Here and in what follows we use the shortened notation g yt (t, s) = g yt (t, s, y 0 (t), y 0 (s), x 0 (s), u 0 (s)), g ys (t, s) = g ys (t, s, y 0 (t), y 0 (s), x 0 (s), u 0 (s)), where g yt and g ys always denote the derivatives with respect to the outer and inner state variables y (i.e., the third and fourth arguments) of g, respectively.
Denote by Ψ the linear operator Lemma 5.2. The operator Ψ is surjective.
Proof. Let us take any z 1 ∈ Z 1 , z 2 ∈ Z 2 , z 3 ∈ Z 3 and try to find w = (y, x, u) ∈ W satisfying (47). We will seek for u in the form is the transposed matrix. Then the third relation in (47), defined by (42), becomes is nondegenerate, and moreover, its inverse (G u (t) G * u (t)) −1 is bounded. Hence it is possible to express v(t) as a function of x(t) and y(t) from the latter relation: Consequently, where Using this expression in the second relation in (47) with account of (41), we get where Note thatz 2 (t 0 ) = 0, i.e.z 2 ∈ Z 2 . Consider the first relation in (47) given in detail by (45). Substituting (49) into (45) and taking into account (48), we obtain Note again thatz 1 (t 0 ) = 0, i.e.z 1 ∈ Z 1 . Setting x(t 0 ) = 0 and y(t 0 ) = 0, we come to a system of the form wherez 1 ∈ Z 1 andz 2 ∈ Z 2 are given functions. Consider the operator Γ : Z 1 × Z 2 → Z 1 × Z 2 mapping a pair (x(t), y(t)) to the pair (z 1 (t),z 2 (t)) defined by (50). According to Lemma 5.1, this operator is surjective. Hence, for any given (z 1 (t),z 2 (t)) in the space Z 1 × Z 2 one can find (x(t), y(t)) in the same space satisfying (50). Defining u(t) as in (48), we obtain a solution to (47).
Since the linear operator T 4 (w 0 ) is finite dimensional, the surjectivity of operator Ψ = (T 1 (w 0 ), T 2 (w 0 ), T 3 (w 0 )) implies that the operator T (w 0 ) = (Ψ, T 4 (w 0 )) has a closed image. Hence the equality constraints of Problem B are weakly regular at the point w 0 .
Observe now that the dual (conjugate) space C * 0 ([t 0 , t 1 ], R n ) to the Banach space C 0 ([t 0 , t 1 ], R n ) consists of all Radon measures having no atoms at zero, i.e., measures defined by functions ψ(t) of bounded variation that are continuous at t = t 0 . Moreover, it is convenient to assume that the functions ψ(t) have a right limit value ψ(t 1 + 0) at the point t 1 , and hence is defined the jump ∆ψ(t 1 ) := ψ(t 1 + 0) − ψ(t 1 − 0) at this point. (The jump at t 0 vanishes since ψ is continuous at t 0 . ) The norm of ψ is its total variation Var ψ.
The elements of conjugate space to Z are the quadruples (ψ x , ψ y , m, β), where ψ x (t) : [t 0 , t 1 ] → R n and ψ y (t) : [t 0 , t 1 ] → R m are functions of bounded variation continuous at t 0 , m ∈ L * ∞ ([t 0 , t 1 ], R d(G) ), and β ∈ R d(η) . It is again convenient to assume that ψ x and ψ y have right limit values ψ x (t 1 + 0) and ψ y (t 1 + 0), and hence there defined the jumps at the point t 1 It is also convenient to assume that the functions ψ x and ψ y are left continuous at any point t ∈ (t 0 , t 1 ]. The inequality constraints. First, let us turn to the state inequality constraints. Note that, for any point ω 0 of the cone Ω = C − ([t 0 , t 1 ], R), an arbitrary element of the supporting cone at this point is given by a nonnegative Riemann-Stieltjes measure dµ(t) generated by a nondecreasing function µ(t) satisfying the condition dµ(t) ω 0 (t) ≡ 0. For any k = 1, . . . , d(Φ), we will use this fact for the function ω 0 (t) = Φ k (t, y 0 (t), x 0 (t)). It implies that the corresponding measure dµ k (t) satisfies the complementary slackness condition dµ k (t) Φ k (t, y 0 (t), x 0 (t)) ≡ 0, i.e. is concentrated on the set Note also, that in view of assumption (12) , the measures dµ k have no atoms at the ends of the interval, i.e. the functions µ k are continuous at t = t 0 and t = t 1 .
Without loss of generality, we can assume that µ k are left continuous on (t 0 , t 1 ), and µ k (t 0 ) = 0. In view of this, Var µ k = µ k (t 1 ).

Now consider the mixed inequality constraints. Note that the description of the dual cone to
, R) is more complicated than to the above cone Ω. We will need the following fact (see, e.g., [14]). Lemma 5.3. Let v 0 ∈ K (i.e., v 0 (t) ≤ 0 a.e.). Then the conditions p ∈ K 0 and p, v 0 = 0 (i.e., p is a support functional to K at the point v 0 ) are equivalent to the conditions: p ≥ 0 and p is concentrated on each of the sets For any i we will apply this lemma to the function v 0 (t) = F i (t, y 0 (t), x 0 (t), u 0 (t)). It implies that the corresponding functional p i is nonnegative and concentrated on the set x 0 (t), u 0 (t)) ≥ −δ } for any δ > 0. Recall that, according to the Yosida-Hewitt theorem [13], any functional p ∈ L * ∞ ([t 0 , t 1 ], R) can be represented in the form p = p a + p s , where p a ∈ L 1 ([t 0 , t 1 ], R) is an absolutely continuous (regular) component, and p s is a singular component. Moreover, ||p|| = ||p a || + ||p s ||. If p ≥ 0, then also p a ≥ 0 and p s ≥ 0.
where χ E k (t) is the characteristic function of the set E k (equal to 1 on E k and 0 outside of E k ). If needed, one can also assume that the sequence E k is decreasing (telescopic), i.e. E k ⊃ E k+1 for all k.
If a functional p is regular and concentrated on each of the sets E k , then obviously it is concentrated on their intersection. For singular functionals, this is not true.
First, let us simplify the notriviality condition.
Note that the functions ψ x and ψ y in the Euler-Lagrange equation (56) are defined up to arbitrary constants. It will be convenient to assume that ψ x (t 1 + 0) = 0, ψ y (t 1 + 0) = 0.
Recall that we assume ψ x and ψ y to be left continuous at any point t ∈ (t 0 , t 1 ], in particular, at the point t 1 . Hence, the jumps of these functions at t 1 are ∆ψ x (t 1 ) = −ψ x (t 1 ), ∆ψ y (t 1 ) = −ψ y (t 1 ).
Recall also that both ψ x and ψ y are continuous at the point t 0 . Condition (56) decomposes into three independent conditions with respect tō x,ȳ andū separately.
Changing the order of integration in the first two summands, we get Define the functions

s). (60)
Here and in what follows we agree for definiteness, that all integrals with respect to measures dψ x , dψ y are taken on the half-open interval [t 0 , t 1 ), and possible jumps of ψ x and ψ y at the point t 1 are selected as separate summands. (As we know, these functions have no jumps at t 0 .) Observe that functions σ x and σ y are measurable and bounded. Equation (58) then becomes Here, all summands can be considered as functionals ofū ∈ L ∞ ([t 0 , t 1 ], R r ). The first of them is absolutely continuous, and the two last ones satisfy the assumption of positive-linear independence (Sec. 2). Our nearest goal is to show that all the functionals h i , m j are also absolutely continuous, i.e., belong to L 1 ([t 0 , t 1 ], R). We will do it similarly to [8].
For any δ > 0, introduce the set of "almost active" indices corresponding to the mixed inequality constraints: where, according to the notation (44), F i (s) = F i (s, y 0 (s), x 0 (s), u 0 (s)). Lemma 6.2. (see [8]). Let assumption RMC be satisfied. Then there exist δ > 0 and c > 0 such that for almost all s ∈ [t 0 , t 1 ], for any numbers α i ≥ 0 and β j , the following estimate holds: Fix some δ > 0 and c > 0 from Lemma 6.2, and recall that each functional h i is concentrated on the set M δ i . Let us split the interval [t 0 , t 1 ] into a finite number of subsets E 1 , . . . , E N of positive measure, on each of which the set of "almost active" indices I k = { i | F i (s) ≥ −δ } is constant. (This can be done, e.g., as follows. Enumerate all possible nonempty subsets of indices contained in {1, . . . , d(F )} and denote them by I k , k = 1, . . . ,k. For each k, let E k be the set of all those s ∈ [t 0 , t 1 ] for which the set of "almost active" indices coincide with I k , i.e., { i | F i (s) ≥ −δ } = I k . Among all E k , select the sets of positive measure. Let these sets be E 1 , . . . , E N . They form a required partition of the interval [t 0 , t 1 ]. ) Now, fix any k ∈ {1, . . . , N } and consider the system of vector-functions Lemma 6.2 implies that, for almost all s ∈ E k , for any numbers α i ≥ 0 and β j , the following estimate holds: In this case, we say that the system of vector-functions p i (s), q j (s) is uniformly positively-linearly independent (UPLI) on the set E k .
Then we use the following theorem, Since this relation holds for allū ∈ L ∞ ([t 0 , t 1 ], R r ), we obtain This is the stationarity condition with respect to u. By Lemma 5.3, each h i is concentrated on the set M 0 i , i.e., the following complementary slackness conditions hold: h i (t) F i (t, y 0 (t), x 0 (t), u 0 (t)) = 0 a.e. on [t 0 , t 1 ], i = 1, . . . , d(F ).
Moreover, setting in (64) α i = h i (s) and β j = m j (s), we obtain for every k the estimate a.e. on E k , whence, the boundedness of σ(s) implies that the functions h i (s), m j (s) are also essentially bounded (not just integrable).
Now, let us take into account that the functions g and h are twice continuously differentiable. In this case, the integrals in (59) and (60) can be taken by parts. Note preliminarily that, for any fixed s, where, like before, the partial derivatives of g and h are evaluated along the reference process w 0 (t) according to (44). Then, using the left-continuity of the functions ψ x , ψ y and the agreement ψ x (t 1 + 0) = ψ y (t 1 + 0) = 0, we can write and similarly, σ y (s) = Let us introduce the function H(s, y, x, u) = ψ x (s) g(s, s, y 0 (s), y, x, u) + ψ y (s) h(s, s, y, u) which we call the modified Pontryagin function. Then condition (70) becomes The x -component of the Euler-Lagrange equation. Now, setȳ = 0 and u = 0 in equality (56).
Changing the order of integration in the second integral, we write In view of the accepted agreement, we represent the integrals with respect to the measure dψ x as follows: Note that the function is measurable and bounded, since the function g x (t, s) is continuous in t and measurable and bounded in s. Consequently, equation (75) becomes Gathering the terms containingx(t 0 ) andx(t 1 ), respectively, and also the terms with the integral ofx(s) over interval (t 0 , t 1 ), we get: Recall that the measures dµ k have no atoms at the points t 0 and t 1 , i.e. the functions µ k are continuous at these points.
Since the functionx(t) on (t 0 , t 1 ) and its valuesx(t 0 ),x(t 1 ) can be chosen "almost independently" of each other (see Lemma A.2 in [8] or Lemma 6.3 in [9]), the coefficients at them must vanish. Thus, we obtain the transversality conditions and the adjoint equation in x as an equality between measures: where ζ(s) is defined in (76). Recall that ∆ψ x (t 1 ) = −ψ x (t 1 ), ψ x (t 1 − 0) = ψ x (t 1 ), and hence the boundary conditions (79) and (80) write: (We also use here the continuity of ψ x at t 0 . ) Now, take again into account that the function g is twice continuously differentiable and rewrite the function ζ, taking by parts the integrals in its definition (76). Note preliminarily that, for any t, s, Then Substituting this expression into (81), we obtain another form of the adjoint equation in x : − dψ x (s) = ψ x (s) g x (s, s) ds Using the generalized derivatives of functions of bounded variation, we can rewrite this equation as or shortly, where H is defined in (73).
Gathering similar terms, we get Changing the order of integration in the double integrals and using the above agreement about the integrals in dψ x , dψ y , we obtain where the function is measurable and bounded (since the functions g ys (t, s) and h y (t, s) are continuous in t and measurable and bounded in s; see (44)).
Problem C. For our main purpose, we will need the following particular case of Problem B in which the function g does not explicitly depend on t, s, nor h on t, s, y. Thus, the control system has the form g(y(t), y(s), x(s), u(s)) ds, h(u(s)) ds.
(99) Then, the expressions for the functions H, H, and R in Problem C are a bit more simple than those in Problem B: H(s, y, x, u) = ψ x (s) g(y 0 (s), y, x, u) + ψ y (s) h(u) Thus, the stationarity conditions for Problem A are in the complete accordance with those for Problem E. The appearance of additional terms in the conditions for ψ t in Problem A is caused by the additional dependence of the control system (129) and the cost (130) in Problem E on the time variable t, since they explicitly involve the function c(t). This time variable appears in Problem A as the outer time and generates the function R.