GENERIC EXISTENCE OF SOLUTIONS OF NONCONVEX OPTIMAL CONTROL PROBLEMS

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In 1996, the author obtained a generic existence and uniqueness result (with respect to variations of the integrand of the integral functional) without the convexity condition for a class of optimal control problems satisfying the Cesari growth condition. In this paper, we survey this result and its recent extensions, and establish several new results in this direction.


Introduction
The Tonelli existence theorem in the calculus of variations [17,18] and its subsequent generalizations and extensions (e.g., [5,11,14,16]) are based on two fundamental hypotheses concerning the behavior of the integrand as a function of the last argument (derivative): one is that the integrand should grow superlinearly at infinity and the other is that it should be convex (or exhibit a more special convexity property in case of a multiple integral with vector-valued functions) with respect to the last variable.Moreover, certain convexity assumptions are also necessary for properties of lower semicontinuity of integral functionals which are crucial in most of the existence proofs, although there are some interesting theorems without convexity (see [5,Chapter 16] and [2,4,13]).
In 1996, the author showed that the convexity condition is not needed generically, and not only for the existence but also for the uniqueness of a solution and even for well-posedness of the problem (with respect to some natural topology in the space of integrands).This result was published in [22].Instead of considering the existence of a solution for a single integrand f , we investigated it for a space of integrands and showed that a unique solution exists for most of the integrands in the space.This approach has already been successfully applied in the theory of dynamical systems (see [6,7,15]), as well as in the calculus of variations (see, e.g., [1,19,21]).Interesting generic existence results were obtained for particular cases of variational problems [3,12].In [3,12] were studied integrands of the form L(x,v) = g(x) + h(v) where h is nonconvex and x is scalarvalued.It was shown in [3] that the set Ᏸ of all continuous functions g such that for any h the corresponding variational problem has a solution is an everywhere dense subset of C(R 1 ) equipped with the topology of uniform convergence on bounded subsets.In [12] it was established that the set Ᏸ is of the first category in C(R 1 ).In [22] the same approach allowed us to establish the generic existence of solutions for a large class of optimal control problems without convexity assumptions.
More precisely, in [22] we considered a class of optimal control problems (with the same system of differential equations, the same functional constraints, and the same boundary conditions) which is identified with the corresponding complete metric space of cost functions (integrands), say Ᏺ.We did not impose any convexity assumptions.These integrands are only assumed to satisfy the Cesari growth condition.The main result in [22] establishes the existence of an everywhere dense G δ -set Ᏺ ⊂ Ᏺ such that for each integrand in Ᏺ , the corresponding optimal control problem has a unique solution.
The next step in this area of research was done in [10].There we introduced a general variational principle having its prototype in the variational principle of Deville et al. [8].A generic existence result in the calculus of variations without convexity assumptions was then obtained as a realization of this variational principle.It was also shown in [10] that some other generic well-posedness results in optimization theory known in the literature and their modifications are obtained as a realization of this variational principle.Note that the generic existence result in [10] was established for variational problems but not for optimal control problems and that the topologies in the spaces of integrands in [10,22] are different.
In [20] we suggested a modification of the variational principle in [10] and applied it to classes of optimal control problems with various topologies in the corresponding spaces of integrands.As a realization of this principle, we established, generic existence results for classes of optimal control problems in which constraint maps are also subject to variations as well as the cost functions.More precisely, we established generic existence results for classes of optimal control problems (with the same system of differential equations, the same boundary conditions, and without convexity assumptions) which are identified with the corresponding complete metric spaces of pairs ( f ,U) (where f is an integrand satisfying the Cesari growth condition and U is a constraint map) endowed with some natural topology.We showed that for a generic pair ( f ,U) the corresponding optimal control problem has a unique solution.
In this paper, we discuss the results of [20,22] and establish extensions of the main result of [20].
Let M denote the set of all (t,x,u) with (t,x) ∈ A, u ∈ U(t,x), and let B 1 ,B 2 ⊂ R n be closed.We assume that the set M is closed and A(t) = ∅ for every t ∈ [T 1 ,T 2 ].Let H(t,x,u) = (H 1 ,...,H n ) be a given continuous function defined on M.
We say that a pair x : [T 1 ,T 2 ] → R n , u : [T 1 ,T 2 ] → R m is admissible if x = (x 1 ,...,x n ) is an absolutely continuous (a.c.) function, u = (u 1 ,...,u m ) is a measurable function, and the following relations hold: Denote by Ω the set of all admissible pairs (x,u).We suppose that Ω = ∅.
In this section, we are concerned with the existence of the minimum in Ω of the functional where h : B 1 × B 2 → R 1 is a lower semicontinuous bounded below function, and f belongs to a space of functions described below.Denote by C l (B 1 × B 2 ) the set of all lower semicontinuous bounded below functions h : B 1 × B 2 → R 1 , and denote by C(B 1 × B 2 ) the set of all continuous functions h ∈ C l (B 1 × B 2 ).For the set C l (B 1 × B 2 ), we consider, the uniformity which is determined by the base where > 0. It is easy to verify that the uniform space C l (B 1 × B 2 ) is metrizable and complete, and C(B 1 × B 2 ) is a closed subset of C l (B 1 × B 2 ).We consider the topological space C(B 1 × B 2 ) ⊂ C l (B 1 × B 2 ) which has the relative topology.Denote by M l the set of all lower semicontinuous functions f : M → R 1 which satisfy the following growth condition.
This growth condition proposed by Cesari (see [5]) and its equivalents and modifications are rather common in the literature.
Denote by M c the set of all continuous functions f ∈ M l .For N, > 0, we set x,u) ≤ + sup f (t,x,u) , g(t,x,u) (t,x,u) ∈ M . (2.5) We can show in a straightforward manner that for the set M l there exists the uniformity which is determined by the base E(N, ), N, > 0. It is easy to verify that the uniform space M l is metrizable and complete.Clearly M c is a closed subset of M l .We consider the topological space M c ⊂ M l which has the relative topology, and the spaces which have the product topology.
We consider the functionals of the form where (x,u For each f ∈ M l , and each h ∈ C l (B 1 × B 2 ), we consider the problem of the absolute minimum and set It is easy to see that (2.10) Denote by mes(E) the Lebesgue measure of a measurable set Denote by Āl,reg the closure of A l,reg in A l , and by Āc,reg the closure of Denote by Mh l,reg the closure of M h l,reg in M l , and by Mh c,reg the closure of M h c,reg in M c .We showed in [22] ) with the relative topology.
In [22] we established the following results which show that generically the optimal control problem considered in this section has a unique solution.
Theorem 2.1.There exist a set F l ⊂ Āl,reg which is a countable intersection of open everywhere dense subsets of Āl,reg , and a set F c ⊂ Āc,reg ∩ F l which is a countable intersection of open everywhere dense subsets of Āc,reg , such that for each ( f ,h) ∈ F l the following assertions hold: (1) µ( f ,h) < ∞ and there exists a unique (x ( f ,h) ,u ( f ,h) ) ∈ Ω for which (2) for each > 0, there exist a neighborhood U of ( f ,h) in A l and a number δ > 0 such that for each (g,ξ) ∈ U and each (x,u) ∈ Ω satisfying I (g,ξ) (x,u) ≤ µ(g,ξ) + δ, the following relation holds: Note that by the Baire category theorem, the set F l is nonempty and in fact everywhere dense in Āl,reg . (2.15)

Optimal control problems with multiple integrals
Let K be a bounded domain in R m where m > 1, let and let W 1,1 0 (K) be the closure of For a function u = (u 1 ,...,u n ), where u i ∈ W 1,1 (K), i = 1,...,n, we set and for every (ω,x) ∈ A, U(ω,x) is a given subset of u-space R N .Let M denote the set of all (ω,x,u) with (ω,x) ∈ A, u ∈ U(ω,x).We assume that the set M is a closed subset of the space K × R n × R N with the product topology.Let H(ω,x,u) be a given continuous function defined on M such that and let θ * = (θ * i ) n i=1 ∈ (W 1,1 (K)) n be fixed.
Denote by M l the set of all lower semicontinuous functions f : M → R 1 which satisfy the following growth condition.
For each > 0 there exists an integrable scalar function Denote by M c the set of all continuous functions f ∈ M l .For N, > 0, we set We can show in a straightforward manner that for the set M l there exists the uniformity which is determined by the base E(N, ), N, > 0. It is easy to verify that the uniform space M l is metrizable and complete.Clearly M c is a closed subset of M l .We consider the topological space M c ⊂ M l which has the relative topology.
We consider the functionals of the form where (x,u) ∈ Ω, f ∈ M l .
For each f ∈ M l , we consider the problem of the absolute minimum and set It is easy to see that In [22] we established the following result which shows that generically the optimal control problem considered in this section has a unique solution.
Theorem 3.1.There exist a set F l ⊂ Ml,reg which is a countable intersection of open everywhere dense subsets of Ml,reg , and a set F c ⊂ Mc,reg ∩ F l which is a countable intersection of open everywhere dense subsets of Mc,reg , such that for each f ∈ F l , the following assertions hold: (1) µ( f ) < ∞ and there is a unique (x ( f ) ,u ( f ) ) ∈ Ω for which for each > 0, there exist a neighborhood U of f in M l and a number δ > 0 such that for each g ∈ U and each (x,u) ∈ Ω satisfying I (g) (x,u) ≤ µ(g) + δ, the following relation holds: (3.12)

Generic well-posedness in nonconvex optimal control
We use the following notations and definitions.Let k ≥ 1 be an integer.We again denote by mes(E) the Lebesgue measure of a measurable set where |α| = k i=1 α i .For each function f : X → [−∞,∞] where X is nonempty, we set inf( f ) = inf{ f (x) : x ∈ X}.For each set-valued mapping U : X → 2 Y \ {∅} where X and Y are nonempty, we set We consider topological spaces with two topologies where one is weaker than the other.(Note that they can coincide.)We refer to them as the weak and the strong topologies, respectively.If (X,d) is a metric space with a metric d and Y ⊂ X, then usually Y is also endowed with the metric d (unless another metric is introduced in Y ).Assume that X 1 and X 2 are topological spaces and that each of them is endowed with a weak and a strong topology.Then for the product X 1 × X 2 , we also introduce a pair of topologies: a weak topology which is the product of the weak topologies of X 1 and X 2 and a strong topology which is the product of the strong topologies of X 1 and X 2 .If Y ⊂ X 1 , then we consider the topological subspace Y with the relative weak and strong topologies (unless other topologies are introduced).If (X i ,d i ), i = 1,2, are metric spaces with the metrics d 1 and d 2 , respectively, then the space X 1 × X 2 is endowed with the metric d defined by Let m,n,N ≥ 1 be integers.We assume that Ω is a fixed bounded domain in R m , H(t,x,u) is a fixed continuous function defined on i=1 and H i = (H i j ) m j=1 , i = 1,...,n, B 1 and B 2 are fixed nonempty closed subsets of R n and θ * = (θ * i ) n i=1 ∈ (W 1,1 (Ω)) n is also fixed.Here and , where C ∞ 0 (Ω) is the space of smooth functions u : Ω → R 1 with compact support in Ω.
If m = 1, then we assume that Ω = (T 1 ,T 2 ), where T 1 and T 2 are fixed real numbers for which T 1 < T 2 .
For a function u = (u 1 ,...,u n ), where u i ∈ W 1,1 (Ω), i = 1,...,n, we set Define set-valued mappings A : For each A : Ω → 2 R n \ {∅} and each U : graph(A) → 2 R N \ {∅} for which graph(U) is a closed subset of the space Ω × R n × R N with the product topology, we denote by X(A,U) the set of all pairs of functions (x,u), where x = (x 1 ,...,x n ) ∈ (W 1,1 (Ω)) n , u = (u 1 ,...,u N ) : Ω → R N is measurable and the following relations hold: Note that in the definition of the space X(A,U) we use the boundary condition (4.7c) in the case m = 1 while in the case m > 1 we use the boundary condition (4.7d).Both of them are common in the literature.We do this to provide a unified treatment for both cases.Note that the main result of the section is valid in the case m = 1 for a class of Bolza problems (with the same boundary condition (4.7c)) while in the case m > 1 it holds for a class of Lagrange problems (with the same boundary condition (4.7d)).
To be more precise, we have to define elements of X(A,U) as classes of pairs equivalent in the sense that (x 1 ,u 1 ) and (x 2 ,u 2 ) are equivalent if and only if x 2 (t) = x 1 (t), u 2 (t) = u 1 (t), t ∈ Ω a.e.If m = 1, then by an appropriate choice of representatives, W 1,1 (T 1 ,T 2 ) can be identified with the set of absolutely continuous functions x : [T 1 ,T 2 ] → R 1 , and we will henceforth assume that this has been done.
Let A : Ω → 2 R n \ {∅}, U : graph(A) → 2 R N \ {∅} and let graph(U) be a closed subset of the space Ω × R n × R N with the product topology.
For the set X(A,U) defined above, we consider the uniformity which is determined by the following base: where > 0. It is easy to see that the uniform space X(A,U) is metrizable (by a metric ρ).
In the space X(A,U) we consider the topology induced by the metric ρ.
Next we define spaces of integrands associated with the maps A and U.By ᏹ(A,U) we denote the set of all functions f : graph(U) → R 1 ∪ {∞} with the following properties: (i) f is measurable with respect to the σ-algebra generated by products of Lebesgue measurable subsets of Ω and Borel subsets of (iii) for each > 0, there exists an integrable scalar function ψ (t) ≥ 0, t ∈ Ω, such that |H(t, x,u)| ≤ ψ (t) + f (t,x,u) for all (t,x,u) ∈ graph(U).Due to the property (i) for every f ∈ ᏹ(A,U) and every (x,u) ∈ X(A,U), the function f (t,x(t),u(t)), t ∈ Ω, is measurable.
Denote by ᏹ l (A,U) (resp., ᏹ c (A,U)) the set of all lower semicontinuous (resp., finitevalued continuous) functions f : graph(U) → R 1 ∪ {∞} in ᏹ(A,U).Now we equip the set ᏹ(A,U) with the strong and weak topologies.For the space ᏹ(A,U), we consider the uniformity determined by the following base: where > 0. It is easy to see that the uniform space ᏹ(A,U) with this uniformity is metrizable (by a metric d ᏹ ) and complete.This uniformity generates in ᏹ(A,U) the strong topology.Clearly ᏹ l (A,U) and ᏹ c (A,U) are closed subsets of ᏹ(A,U) with this topology.
For each > 0, we set , and for a.e.t ∈ Ω, for each x ∈ A(t), each u ∈ U(t,x) . (4.10) Using the following simple lemma, we can easily show that for the set ᏹ(A,U) there exists the uniformity which is determined by the base E ᏹw ( ), > 0. This uniformity induces in ᏹ(A,U) the weak topology.
Denote by C l (B 1 × B 2 ) the set of all lower semicontinuous functions ξ : B 1 × B 2 → R 1 ∪ {∞} bounded from below.We also equip the set C l (B 1 × B 2 ) with strong and weak topologies.For the set C l (B 1 × B 2 ), we consider the uniformity determined by the following base: where > 0. It is easy to see that the uniform space C l (B 1 × B 2 ) is metrizable (by a metric d c ) and complete.This metric induces in C l (B 1 × B 2 ) the strong topology.
For any > 0, we set where > 0. By using Lemma 4.1, we can easily show that for the set C l (B 1 × B 2 ) there exists a uniformity which is determined by the base E cw ( ), > 0. This uniformity induces in C l (B 1 × B 2 ) the weak topology.Denote by with the weak topology.
In the case m > 1 for each f ∈ ᏹ(A,U) we define In the case m=1 for each f ∈ᏹ(A, U) and each ξ ∈ C l (B 1 × B 2 ) we define We showed (see [20, Propositions 4.1 and 4.2]) that in both cases (4.15) and (4.16) define lower semicontinuous functionals on X(A,U).
From now on in this section, we consider a fixed set-valued mapping A : Ω → 2 R n \ {∅} for which graph(A) is a closed subset of the space Ω × R n with the product topology.Denote by U A the restriction of U (see (4.6)) to the graph(A).Namely, We consider functionals ) and functionals I ( f ) with f ∈ ᏹ(A, U A ) (in the case m > 1) defined on the space X(A, U A ) (see (4.7)).The main result of this section is established for several classes of optimal control problems with different corresponding spaces of the integrands which are subsets of the space ᏹ(A, U A ).The subspaces of lower semicontinuous and continuous integrands (ᏹ l (A, U A ) and ᏹ c (A, U A )) have already been defined.Now we define subspaces of ᏹ(A, U A ) which consist of integrands differentiable with respect to the control variable u.
Let k ≥ 1 be an integer.Denote by ᏹ k (A, U A ) the set of all finite-valued f ∈ ᏹ(A, U A ) such that for each (t,x) ∈ graph(A) the function f (t,x,•) ∈ C k (R N ).We consider the topological subspace ᏹ k (A, U A ) ⊂ ᏹ(A, U A ) with the relative weak topology.The strong topology on ᏹ k (A, U A ) is induced by the uniformity which is determined by the following base: where > 0. It is easy to see that the space ᏹ k (A, U A ) with this uniformity is metrizable (by a metric d ᏹ,k ) and complete.Define with the strong topology.Finally we define subspaces of ᏹ( A, U) which consist of integrands differentiable with respect to the state variable x and the control variable u.Denote by ) with the relative weak topology.The strong topology in ᏹ * k ( A, U) is induced by the uniformity which is determined by the following base: where > 0. It is easy to see that the space ᏹ * k ( A, U) with this uniformity is metrizable (by a metric d * ᏹ,k ) and complete.Define with the strong topology.Thus we have defined all the spaces of integrands for which we will state our main result of this section.Now we will define a space of constraint maps ᏼ A .Denote by S(R N ) the set of all nonempty convex closed subsets of R N .For each x ∈ R N and each is the Hausdorff distance between C 1 and C 2 .For the space S(R N ), we consider the uniformity determined by the following base: where > 0. It is well known that the space S(R N ) with this uniformity is metrizable and complete.Denote by ᏼ A the set of all set-valued mappings U : graph(A) → S(R N ) such that graph(U) is a closed subset of the space graph(A) × R N with the product topology.
For the space ᏼ A , we consider the uniformity determined by the following base: where > 0. It is easy to see that the space ᏼ A with this uniformity is metrizable and complete.
We consider the space X(A, U A ) with the metric ρ (see (4.8)).For each U ∈ ᏼ A , define

.25)
In the case m = 1 for each we consider the optimal control problem and in the case m > 1 for each U ∈ ᏼ A and each f ∈ ᏹ(A, U A ) we consider the optimal control problem We will state the main result of this section, Theorem 4.2, in such a manner that it will be applicable to the Bolza problem in case m = 1 and to the Lagrange problem in case m > 1, and also applicable for all the spaces of integrands defined above.
To meet this goal, we set Ꮽ 2 = ᏼ A and define a space Ꮽ 1 as follows: where , and Ꮽ 11 is one of the following spaces:

is an integer and
In [20], we showed that J a is lower semicontinuous for all a ∈ Ꮽ 1 × A 2 .Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 × Ꮽ 2 : inf(J a ) < ∞} in the space Ꮽ 1 × A 2 with the strong topology.We assume that Ꮽ is nonempty.The following theorem established in [20] is the main result of this section.
Theorem 4.2.There exists an everywhere dense (in the strong topology) set Ꮾ ⊂ Ꮽ which is a countable intersection of open (in the weak topology) subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf(J a ) is finite and attained at a unique pair (x, ū) ∈ X(A, U A ), (2) for each > 0 there are a neighborhood ᐂ of a in Ꮽ with the weak topology and

Generic variational principle
Theorem 4.2 is obtained as a realization of a variational principle which was introduced in [20].This variational principle is a modification of the variational principle in [10].
We consider a metric space (X,ρ) which is called the domain space and a complete metric space (Ꮽ,d) which is called the data space.We always consider the set X with the topology generated by the metric ρ.For the space Ꮽ, we consider the topology generated by the metric d.This topology will be called the strong topology.In addition to the strong topology, we also consider a weaker topology on Ꮽ which is not necessarily Hausdorff.This topology will be called the weak topology.(Note that these topologies can coincide.)We assume that with every a ∈ Ꮽ a lower semicontinuous function f a on X is associated with values in R = [−∞,∞].In our study, we use the following basic hypotheses about the functions.
(H1) For any a ∈ Ꮽ, any > 0, and any γ > 0, there exist a nonempty open set ᐃ in Ꮽ with the weak topology, x ∈ X, α ∈ R 1 , and η > 0 such that and for any b ∈ ᐃ, n=1 ⊂ X is a Cauchy sequence, and the sequence In [20] we showed (see Theorem 5.1 below) that if (H1) and (H2) hold, then for a generic a ∈ Ꮽ the minimization problem f a (x) → min, x ∈ X, has a unique solution.This result generalizes the variational principle in [10, Theorem 2.2] which was obtained for the complete domain space (X,ρ).Note that if (X,ρ) is complete, the weak and strong topologies on Ꮽ coincide and for any a ∈ Ꮽ the function f a is not identically ∞, then the variational principles in [10] and in this section are equivalent.
For the classes of optimal control problems considered in this paper, the domain space is usually the space X(A, U A ) with the metric ρ (see (4.8)) which is not complete.Since the variational principle in [10] was established only for complete domain spaces, it cannot be applied to these classes of optimal control problems.Fortunately, instead of the completeness assumption, we can use (H2) and this hypothesis holds for spaces of integrands (integrand-map pairs) which satisfy the Cesari growth condition.In [20] we established the following result.
Theorem 5.1.Assume that (H1) and (H2) hold.Then there exists an everywhere dense (in the strong topology) set Ꮾ ⊂ Ꮽ which is a countable intersection of open (in the weak topology) subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf( f a ) is finite and attained at a unique point x ∈ X, (2) for each > 0, there are a neighborhood ᐂ of a in Ꮽ with the weak topology and δ > 0 Following the tradition, we can summarize the theorem by saying that under the assumptions (H1) and (H2) the minimization problem for f a on (X,ρ) is generically strongly well-posed with respect to Ꮽ.
The proof of Theorem 4.2 consists in verifying that hypotheses (H1) and (H2) hold for the space of integrand-map pairs introduced in Section 4. To simplify the verification of (H1) in [20] we introduced new assumptions (A1)-(A4) and showed that they imply (H1) (see Proposition 5.3 below).
Let (X,ρ) be a metric space with the topology generated by the metric ρ and let (Ꮽ 1 ,d 1 ), (Ꮽ 2 ,d 2 ) be metric spaces.For the space Ꮽ i (i = 1,2), we consider the topology generated by the metric d i .This topology is called the strong topology.In addition to the strong topology we consider a weak topology on Assume that with every a ∈ Ꮽ 1 a lower semicontinuous function φ a : Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 × Ꮽ 2 : inf( f a ) < ∞} in the space Ꮽ 1 × Ꮽ 2 with the strong topology.We assume that Ꮽ is nonempty.We use the following hypotheses.(A1) For each a 1 ∈ Ꮽ 1 , inf(φ a1 ) > −∞ and for each a ∈ Ꮽ 1 × Ꮽ 2 , the function f a is lower semicontinuous.
For each γ ∈ (0,1), each a ∈ Ꮽ 1 , each nonempty set Y ⊂ X, and each x ∈ Y for which there is ā ∈ Ꮽ 1 such that the following conditions hold: for each y ∈ Y satisfying the inequality ρ(y, x) ≤ γ is valid.
The following result was established in [20].
This implies that there exists an open set Ᏺ in Ꮽ 1 × Ꮽ 2 with the weak topology such that inf( f a ) < ∞ for all a ∈ Ᏺ and Ꮽ is the closure of Ᏺ in the space Ꮽ 1 × Ꮽ 2 with the strong topology.

Preliminary results for hypotheses (A2) and (H2)
In this section, we present several auxiliary results obtained in [20].Assume that A : and that graph(U) is a closed subset of the space Ω × R n × R N with the product topology.Consider the spaces X(A,U), ᏹ(A,U), and C l (B 1 × B 2 ) introduced in Section 4.
The following proposition is an auxiliary result for hypothesis (H2).
with the weak topology such that for each ξ ∈ ᐂ and each The following proposition is an auxiliary result for assumption (A2).Proposition 6.5.Let f ∈ ᏹ(A,U) and ∈ (0,1), D > 0. Then there exists a neighborhood ᐂ of f in ᏹ(A,U) with the weak topology such that for each g ∈ ᐂ and each (x,u) ∈ X(A,U) satisfying the following relation holds: Corollary 6.6.Let f ∈ ᏹ(A,U) and > 0. Then there exists a neighborhood ᐂ of f in ᏹ(A,U) with the weak topology such that for all g ∈ ᐂ, inf , and ∈ (0,1), D > 0. Then there exist a neighborhood ᐁ of f in ᏹ(A,U) with the weak topology and a neighborhood ᐂ of h in C l (B 1 × B 2 ) with the weak topology such that for each (ξ,g) ∈ ᐂ × ᐁ and each (x,u) ∈ X(A,U) which satisfies min I ( f ,h) (x,u),I (g,ξ) (x,u) ≤ D, (6.5) the following relations are valid:

Preliminary lemma for hypothesis (A3)
Fix a number d 0 ∈ (0,1).There is a C ∞ -function Now we define a set ᏸ ⊂ C l (B 1 × B 2 ).In the case m = 1 we set ᏸ = C l (B 1 × B 2 ) and in the case m > 1 denote by ᏸ a singleton {0} where 0 is a function in C l (B 1 × B 2 ) which is identical zero.In the case m > 1 for each ( f ,ξ) ∈ ᏹ(A,U) × ᏸ and each (x,u) ∈ X(A,U) we set (see (4.15) and (4.16)).For each measurable set E ⊂ R m , each measurable set E 0 ⊂ E, and each h ∈ L 1 (E), we set Fix an integer k ≥ 1.It is easy to verify that all partial derivatives of the functions (x, y) → φ(|x − y| 2 ), (x, y) ∈ R q × R q with q = n, N up to the order k, are bounded (by some d > 0).
Then there is g : such that for a function f ∈ ᏹ(A,U) defined by f (t,x,u) = f (t,x,u) + g(t,x,u), (t,x,u) ∈ graph(U), (7.9) the following properties hold: for each (y,v) ∈ Y satisfying the relation ρ((y,v),(x, ū)) ≤ γ is valid.Moreover, the function g is the sum of two functions, one of them depending only on (t,x) while the other depending only on (t,u).
The following result was established in [20].
and let ,δ > 0. Then there are U * ∈ ᏼ A , (x, ū) ∈ X(A,U * ), and an open set ᐃ in ᏼ A such that and for all V ∈ ᐃ, with the product topology.We consider the metric space X(A,U) with the metric ρ (see (4.8)).
Now we define Ꮽ 1 as follows: where , and Ꮽ 11 is one of the following spaces: (here k ≥ 1 is an integer, U = U A , and graph(A) is a closed subset of the space Ω × R n with the product topology),

is an integer and
Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 : inf(I (a) ) < ∞} in the space Ꮽ 1 with the strong topology.We assume that Ꮽ is nonempty.The following result is proved analogously to Theorem 4.2.
Theorem 9.1.There exists an everywhere dense (in the strong topology) set Ꮾ ⊂ Ꮽ which is a countable intersection of open (in the weak topology) subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf(I (a) ) is finite and attained at a unique pair (x, ū) ∈ X(A,U), (2) for each > 0, there are a neighborhood ᐂ of a in Ꮽ with the weak topology and In the sequel, we use the notation and definitions from Sections 4 and 5. Let m,n,N ≥ 1 be integers.We again assume that Ω is a fixed bounded domain in R m , H(t,x,u) is a fixed continuous function defined on i=1 and H i = (H i j ) m j=1 , i = 1,...,n, B 1 and B 2 are fixed nonempty closed subsets of R n and θ * = (θ * i ) n i=1 ∈ (W 1,1 (Ω)) n is also fixed.If m = 1, then we assume that Ω = (T 1 ,T 2 ), where T 1 and T 2 are fixed real numbers for which T 1 < T 2 .
Define set-valued mappings A : Consider the metric space (X( A, U),ρ) (see (4.7)) and the spaces of integrands defined in Section 4.
Denote by S(R n × R N ) the set of all nonempty convex closed subsets of is the Hausdorff distance between C 1 and C 2 .For the space S(R n × R N ), we consider the uniformity determined by the following base: where > 0. It is well known that the space S(R n × R N ) with this uniformity is metrizable and complete (see Section 4).Denote by ᏼ the set of all set-valued mappings Q : and a set-valued mapping For the space ᏼ, we consider the uniformity determined by the following base: where > 0. It is not difficult to verify that the space ᏼ with this uniformity is metrizable and complete.We equip the set ᏼ with the topology induced by this uniformity.
For each Q ∈ ᏼ, define In the case m = 1 for each Q ∈ ᏼ and each ( f ,ξ) ∈ ᏹ( A, U) × C l (B 1 × B 2 ) we consider the optimal control problem and in the case m > 1 for each Q ∈ ᏼ and each f ∈ ᏹ( A, U) we consider the optimal control problem We set Ꮽ 2 = ᏼ and define a space Ꮽ 1 as follows: where , and Ꮽ 11 is one of the following spaces: By Propositions 6.1 and 6.2, J a is lower semicontinuous for all a ∈ Ꮽ 1 × A 2 .Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 × Ꮽ 2 : inf(J a ) < ∞} in the space Ꮽ 1 × A 2 with the strong topology.We assume that Ꮽ is nonempty.The following result is an extension of Theorem 4.2.
Theorem 9.2.There exists an everywhere dense (in the strong topology) set Ꮾ ⊂ Ꮽ which is a countable intersection of open (in the weak topology) subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf(J a ) is finite and attained at a unique pair (x, ū) ∈ X( A, U), (2) for each > 0, there are a neighborhood ᐂ of a in Ꮽ with the weak topology and δ > 0 such that for each b ∈ ᐂ, inf(J b ) is finite and if (z,w) ∈ X( A, U) satisfies J b (z,w) ≤ inf(J b ) + δ, then ρ((x, ū),(z,w)) ≤ and |J b (z,w) − J a (x, ū)| ≤ .
Proof of Theorem 9.2.By Propositions 6.1 and 6.2 (A1) holds and J a is lower semicontinuous for all a ∈ Ꮽ 1 × Ꮽ 2 .By Theorem 5.1 we need to verify that (H1) and (H2) are valid.Hypothesis (H2) follows from Proposition 6.2.Therefore it is sufficient to show that (H1) holds.By Proposition 5.3 it is sufficient to show that (A2), (A3), and (A4) are valid.Hypothesis (A2) follows from Propositions 6.5 and 6.7.By Lemma 7.1 (A3) holds.It is easy to see that (A4) follows from Lemma 9.3 proved below.Its proof is a modification of the proof of Lemma 8.2.

An extension of Theorem 4.2
In this section, we use the notation and definitions from Sections 4 and 5.
Let m = 1 and let n,N ≥ 1 be integers, B 1 , B 2 = R n , Ω = (T 1 ,T 2 ), where T 1 and T 2 are fixed real numbers for which T 1 < T 2 , and let H(t,x,u) be a fixed continuous function defined on i=1 .Consider a fixed set-valued mapping A : Ω → 2 R n \ {∅} for which graph(A) is a closed subset of the space Ω × R n with the product topology and a set-valued mapping U A : graph(A) → 2 R N defined by (see (4.17)).We consider the metric space X(A, U A ) with the metric ρ (see (4.8)), the uniform space ᏼ A , and the space of integrands ᏹ(A, U A ) and all its subspaces introduced in Section 4. Note that all of these spaces are equipped with the corresponding uniformities and topologies introduced in Section 4.
Denote by S(R n ) the set of all nonempty convex closed subsets of R n .For each x ∈ R n and each is the Hausdorff distance between C 1 and C 2 .For the space S(R n ), we consider the uniformity determined by the following base: where > 0. It is well known that the space S(R n ) with this uniformity is metrizable and complete. For we consider the optimal control problem A and define a space Ꮽ 1 as follows.
, and Ꮽ 11 is one of the following spaces:

is an integer and A = A).
(10.6) For each a = (a 1 ,a 2 ) ∈ Ꮽ 1 × Ꮽ 2 , we define J a : By Propositions 6.1 and 6.2 J a is lower semicontinuous for all a ∈ Ꮽ 1 × Ꮽ 2 .Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 × Ꮽ 2 : inf(J a ) < ∞} in the space Ꮽ 1 × Ꮽ 2 with the strong topology.We assume that Ꮽ is nonempty.We prove the following result.

A class of nonconvex optimal control problems
In this section, we again use the notation and definitions from Sections 4 and 5. Let m,n,N ≥ 1 be integers, B 1 and B 2 fixed nonempty closed subsets of R n , Ω a fixed bounded domain in R m , H(t,x,u) be a fixed continuous function defined on Consider a fixed set-valued mapping A : Ω → 2 R n \ {∅} for which graph(A) is a closed subset of the space Ω × R n with the product topology and a set-valued mapping U : graph(A) → 2 R N defined by (see (4.17)).We consider the metric space X(A, U A ), the uniform space ᏼ A , the space of integrands ᏹ(A, U A ) and its subspaces introduced in Section 4. Note that all these spaces are equipped with corresponding uniformities and topologies introduced in Section 4.
Denote by M the set of all functions f : graph(A) × R N → R 1 ∪ {∞} with the following properties: (a) f is measurable with respect to the σ-algebra generated by products of Lebesgue measurable subsets of Ω and Borel subsets of For the space M, we consider the uniformity determined by the following base: where > 0. It is easy to see that the uniform space M with this uniformity is metrizable (by a metric d M ) and complete.This uniformity generates in M the strong topology.Denote by M l (resp., M c ) the set of all lower semicontinuous (resp., finite-valued continuous) functions f : graph(A) × R N → R 1 ∪ {∞}.Clearly M l and M c are closed subsets of M with the strong topology.It is easy to see that ᏹ(A, U A ) is a closed subset of M with the strong topology.
For each > 0, we set Using Lemma 4.1 we can easily show that for the set M there exists the uniformity which is determined by the base E M ( ), > 0. This uniformity induces in M the weak topology.Analogously to Proposition 6.1 we can prove the following result.
Analogously to Proposition 6.5 we can prove the following result.
Proposition 11.2.Let f ∈ M and ∈ (0,1), D > 0. Then there exists a neighborhood ᐂ of f in M with the weak topology such that for each g ∈ ᐂ and each (x,u) ∈ X(A, U A ) satisfying the following relation holds: Denote by the of all functions ξ : graph(A) → (−∞,∞] such that for a.e.t ∈ Ω the function ξ(t,•) : A(t) → (−∞,∞] is lower semicontinuous.For the set , we consider the uniformity determined by the following base: where > 0. It is easy to see that the space with this uniformity is metrizable (by a metric d ) and complete.This uniformity generates in the strong topology.
For each > 0, we set where > 0. Using Lemma 4.1 we can easily show that for the set there exists the uniformity which is determined by the base E Mw ( ), > 0. This uniformity induces in the weak topology.Denote by l (resp., c ) the set of all lower semicontinuous (resp., finitevalued continuous) functions ξ ∈ .Clearly l and c are closed subsets of with the strong topology.
Let k ≥ 1 be an integer.In the case and each ψ ∈ , we consider the optimal control problem In the case m > 1 for each f ∈ ᏹ(A, U A ), each U ∈ ᏼ A , each ψ 1 ,...,ψ k ∈ M, and each ψ ∈ , we consider the optimal control problem Define a space Ꮽ 1 as follows: where , and Ꮽ 11 is one of the following spaces:

is an integer and
(11.12) Define a space Ꮽ 2 as follows: where Ꮽ 20 is either or l or c and For each a = (a 0 ,...,a k ,U) ∈ Ꮽ 2 , define and For each a = (a 1 ,a 2 ) ∈ Ꮽ, we define J a : J a (x,u) = ∞, (x,u) ∈ X A, U A \ S a2 .(11.16)By Propositions 6.1, 6.2, and 11.1 J a is lower semicontinuous function for all a ∈ Ꮽ 1 × Ꮽ 2 .
Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 × Ꮽ 2 : inf(J a ) < ∞} in the space Ꮽ 1 × A 2 with the strong topology.We assume that Ꮽ is nonempty.We will establish the following result.
Theorem 11.3.There exists an everywhere dense (in the strong topology) set Ꮾ ⊂ Ꮽ which is a countable intersection of open (in the weak topology) subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf(J a ) is finite and attained at a unique pair (x, ū) ∈ X(A, U A ), (2) for each > 0, there are a neighborhood ᐂ of a in Ꮽ with the weak topology and Proof.By Propositions 6.1, 6.2, and 11.1 (A1) holds and J a is lower semicontinuous for all a ∈ Ꮽ 1 × Ꮽ 2 .By Theorem 5.1 we need to verify that (H1) and (H2) are valid.Hypothesis (H2) follows from Proposition 6.2.Therefore it is sufficient to show that (H1) holds.By Proposition 5.3 it is sufficient to show that (A2), (A3), and (A4) are valid.Hypothesis (A2) follows from Propositions 6.5 and 6.7.By Lemma 7.1, (A3) holds.Hypothesis (A4) will follow from Lemma 11.4 below.

Minimization problems with constraints
In this section, we discuss three classes of minimization problems with constraints.For these classes, generic existence of solutions is obtained as a realization of our variational principle (see Theorem 5.1 and Proposition 5.3).Let (X,ρ) be a complete metric space and let C l (X) be the set of all lower semicontinuous functions f : X → R 1 ∪ {∞}.Denote by C bl the set of all bounded from bellow functions f ∈ C l (X).
For each function f where Y is nonempty set, we define We use the convention that ∞ − ∞ = 0. Denote by C(X) the set of all continuous real-valued functions f ∈ C l (X) and set C b (X) = C(X) ∩ C bl (X).We equip the set C l (X) with a strong and weak topologies.
For the set C l (X), we consider the uniformity determined by the following base: where > 0. Clearly this uniform space C l (X) is metrizable (by a metric d Cs ) and complete.We equip the set C l (X) with the strong topology induced by this uniformity.Now we equip the set C l (X) with a weak topology.For each > 0, we set We can show in a straightforward manner that for the set C l (X) there exists a uniformity which is determined by the base E Cw ( ), > 0. It is easy to see that this uniformity is metrizable (by a metric d Cw ) and complete.This uniformity induces on C l (X) the weak topology.Clearly C(X), C b (X), and C bl (X) are closed subsets of C l (X) with the strong topology.Now we define spaces Ꮽ 1 and Ꮽ 2 .Let Ꮽ 1 be either C bl (X) or C b (X) and let ..,n is one of the following spaces: For a ∈ Ꮽ 1 , we set φ a = a and for g = (g 1 ,...,g n ) ∈ Ꮽ 2 , we set Denote by Ꮽ the closure of the set {a ∈ Ꮽ 1 × Ꮽ 2 : inf( f a ) < ∞} in the space Ꮽ 1 × Ꮽ 2 with the strong topology.
The following result was established in [23].
Theorem 12.1.There exists an everywhere dense (in the strong topology) set Ꮾ ⊂ Ꮽ which is a countable intersection of open (in the weak topology) subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf( f a ) is finite and attained at a unique point x ∈ X; (2) for each > 0, there are a neighborhood ᐂ of a in Ꮽ with the weak topology and δ > 0 such that for each b Note that an analogous result was established in [9] when X is a Banach space and constraint functions are convex.Now we present the second main result of [23].
Let (X, • ) be a Banach space.Consider the set ᏸ of all bounded from below lower semicontinuous functions f : X → R 1 .For the set ᏸ, we consider the uniformity determined by the following base: where > 0. Clearly this uniform space is metrizable and complete.We equip the space ᏸ with the topology induced by this uniformity.For x ∈ X and A ⊂ X, set Denote by S(X) the set of all nonempty closed convex subsets of X.For the set S(X), we consider the uniformity determined by the following base: where > 0. It is well known that the space S(X) with this uniformity is metrizable (by a metric H) and complete.We consider the set S(X) endowed with the Hausdorff topology induced by this uniformity.Set (12.10) Clearly inf( f a ) is finite for all a ∈ Ꮽ.
The following result was established in [23].
Theorem 12.2.There exists an everywhere dense set Ꮾ ⊂ Ꮽ which is a countable intersection of open everywhere dense subsets of Ꮽ such that for any a ∈ Ꮾ, the following assertions hold: (1) inf( f a ) is finite and attained at a unique point x ∈ X; (2) for each > 0 there are a neighborhood ᐂ of a in Ꮽ and δ > 0 such that for each Let (X, • ) be a Banach space, ρ(x, y) = x − y , x, y ∈ X, (12.11) and let n ≥ 1 be an integer.We consider the minimization problem where f ∈ C bl (X), g i ∈ C l (X), i = 1,...,n, A ∈ S(X).
Denote by Ꮽ the closure of the set {a ∈ Ꮽ : inf( f a ) < ∞} in the strong topology.We assume that Ꮽ = ∅.In this section, we establish, the following result.
In the sequel we need the following auxiliary result (see [23,Proposition 7.1]).
Hypothesis (A4) will follows from the next lemma.

( 2 )
for each > 0, there exist δ > 0 and a neighborhood ᐂ of a in Ꮽ with the weak topology such that for each b∈ ᐂ, inf( f b ) is finite and if z ∈ X satisfies f b (z) ≤ inf( f b ) + δ, then z − x a ≤ and | f b (z) − f a (x a )| ≤ .
Theorem 2.2.Let η ∈ C l (B 1 × B 2 ) be fixed and let F l , F c be as guaranteed in Theorem 2.1.Then there exist a set F η l which is a countable intersection of open everywhere dense subsets of Mη c,reg, such that .11) Denote by Ml,reg the closure of M l,reg in M l , and by Mc,reg the closure of M c,reg in M c .The set M l,reg is an open subset of M l , and a set M c,reg is an open subset of M [22,ee[22,  Lemma 7.2]).We consider the topological subspaces Ml,reg , Mc,reg which have the relative topology.