DERIVED CONES TO REACHABLE SETS OF A CLASS OF SECOND-ORDER DIFFERENTIAL INCLUSIONS

DERIVED CONES TO REACHABLE SETS OF A CLASS OF SECOND-ORDER DIFFERENTIAL INCLUSIONS AURELIAN CERNEA We consider a second-order differential inclusion and we prove that the reachable set of a certain second-order variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. This result allows to obtain a sufficient condition for local controllability along a reference trajectory. AMS 21 Subject Classification: 34A6, 93C15. Key words: derived cone, cosine family of operators, mild solution, local controllability. 1. INTRODUCTION The concept of derived cone to an arbitrary subset of a normed space has been introduced by M. Hestenes in [15] and successfully used to obtain necessary optimality conditions in Control Theory. Afterwards, this concept has been largely ignored in favor of other concepts of tangents cones, that may intrinsically be associated to a point of a given set: the cone of interior directions, the contingent, the quasitangent and, above all, Clarke s tangent cone e.g., [1]). Mirică [16, 17]) obtained an intersection propert of derived cones that allowed a conceptually simple proof and significant extensions of the maximum principle in optimal control; moreover, other properties of derived cones may be used to obtain controllability and other results in the qualitative theory of control systems. In our previous papers [4 6, 13] we identified derived cones to the reachable sets of certain classes of discrete and differential inclusions in terms of a variational inclusion associated to the initial discrete or differential inclusion. In the present paper we consider second-order differential inclusions of the form 1.1) x Ax + F t, x), x) X, x ) X 1, where F : [,T] X PX) is a set-valued map, A is the infinitesimal generator of a strongly continuous cosine family of operators {Ct); t R} REV. ROUMAINE MATH. PURES APPL., 56 211), 1, 1 11

2 Aurelian Cernea 2 on a separable Banach space X and X,X 1 X are closed sets. Our aim is to prove that the reachable set of a certain second-order variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the problem 1.1). In order to obtain the continuity property in the definition of a derived cone we shall use a continuous version of Filippov s theorem for mild solutions of differential inclusions 1.1) obtained in [8]. As an application, when X is finite dimensional, we obtain a sufficient condition for local controllability along a reference trajectory. We note that existence results and qualitative properties of the mild solutions of problem 1.1) may be found in [2, 3, 7 12] etc. The paper is organized as follows: in Section 2 we present the notations and the preliminary results to be used in the sequel and in Section 3 we provide our main results. 2. PRELIMINARIES Since the reachable set to a control system is, generally, neither a differentiable manifold, nor a convex set, its infinitesimal properties may be characterized only by tangent cones in a generalized sense, extending the classical concepts of tangent cones in Differential Geometry and Convex Analysis, respectively. From the rather large number of convex approximations, tents, regular tangents cones, etc. in the literature, we choose the concept of derived cone introduced by M. Hestenes in [15]. Let X, ) be a normed space. Definition 2.1 [15]). A subset M X is said to be a derived set to E X at x E if for any finite subset {v 1,...,v k } M, there exist s > and a continuous mapping a ) :[,s ] k E such that a) = x and a ) is conically) differentiable at s = with the derivative col[v 1,...,v k ]inthe sense that aθ) a) k i=1 lim θ iv i =. R k + θ θ We shall write in this case that the derivative of a ) ats = is given by k Da)θ = θ j v j θ =θ 1,...,θ k ) R k + := [, ) k. i=1 AsubsetC X is said to be a derived cone of E at x if it is a derived set and also a convex cone. For the basic properties of derived sets and cones we refer to M. Hestenes [15]; we recall that if M is a derived set then M {} as well as the convex

3 Derived cones to reachable sets of a class of second-order differential inclusions 3 cone generated by M, definedby { k } ccom) = λ j v j ; λ j, k N, v j M, j =1,...,k i=1 is also a derived set, hence a derived cone. The fact that the derived cone is a proper generalization of the classical concepts in Differential Geometry and Convex Analysis is illustrated by the following results [15]): if E R n is a differentiable manifold and T x E is the tangent space in the sense of Differential Geometry to E at x T x E = {v R n ; c : s, s) E, of class C 1,c) = x, c ) = v}, then T x E is a derived cone; also, if E R n is a convex subset then the tangent cone in the sense of Convex Analysis defined by TC x E =cl{ty x); t, y E} is also a derived cone. Since any convex subcone of a derived cone is also a derived cone, such an object may not be uniquely associated to a point x E; moreover, simple examples show that even a maximal with respect to set-inclusion derived cone may not be uniquely defined: if the set E R 2 is defined by E = C 1 C 2, C 1 = {x, x); x }, C 2 = {x, x), x }, then C 1 and C 2 are both maximal derived cones of E at the point, ) E. On the other hand, the up-to-date experience in Nonsmooth Analysis shows that for some problems, the use of one of the intrinsic tangent cones may be preferable. From the multitude of the intrinsic tangent cones in the literature e.g. [1]), the contingent, the quasitangent intermediate) and Clarke s tangent cones, defined, respectively, by { } K x E = v X; s m +, x m x, x m E : xm x s m v, { } Q x E = C x E = v X; s m +, x m x, x m E : xm x s m v { v X; x m,s m ) x, +), x m E, y m E :, y m x m s m } v seem to be among the most oftenly used in the study of different problems involving nonsmooth sets and mappings. An outstanding property of a derived cone, obtained by Hestenes [15], Theorem 4.7.4) is stated in the next lemma. Lemma 2.2 [15]). Let X = R n.thenx inte) if and only if C = R n is a derived cone at x E to E.

4 Aurelian Cernea 4 Corresponding to each type of tangent cone, say τ x E, one may introduce e.g. [1]) a set-valued directional derivative of a multifunction G ) : E X PX) in particular of a single-valued mapping) at a point x, y) GraphG) as follows τ y Gx; v) ={w X; v, w) τ x,y) GraphG)}, v τ x E. We recall that a set-valued map, A ) : X PX) issaidtobea convex respectively, closed convex) process if GraphA )) X X is a convex respectively, closed convex) cone. For the basic properties of convex processeswereferto[1],butweshallusehereonlytheabovedefinition. LetusdenotebyItheinterval[,T]andletX be a real separable Banach space with the norm and with the corresponding metric d, ). Denote by LI)theσ-algebra of all Lebesgue measurable subsets of I, bypx) the family of all nonempty subsets of X and by BX) the family of all Borel subsets of X. Recall that the Pompeiu-Hausdorff distance of the closed subsets A, B X is defined by d H A, B) =max{d A, B), d B,A)}, d A, B) =sup{da, B); a A}, where dx, B) =inf y B dx, y). As usual, we denote by CI,X) the Banach space of all continuous functions x ) :I X endowed with the norm x ) C =sup t I xt) and by L 1 I,X) the Banach space of all Bochner) integrable functions x ) :I X endowed with the norm x ) 1 = I xt) dt. We recall that a family {Ct); t R} of bounded linear operators from X into X is a strongly continuous cosine family if the following conditions are satisfied i) C) = Id, where Id is the identity operator in X, ii) Ct + s)+ct s) =2Ct)Cs) t, s R, iii) the map t Ct)x is strongly continuous x X. The strongly continuous sine family {St); t R} associated to a strongly continuous cosine family {Ct); t R} is defined by St)y := t Cs)yds, y X, t R. The infinitesimal generator A : X X of a cosine family {Ct); t R} is defined by d 2 ) Ay = dt 2 Ct)y. t= For more details on strongly continuous cosine and sine families of operators we refer to [14, 18]. In what follows A is infinitesimal generator of a cosine family {Ct); t R} and F, ) :I X PX) isaset-valuedmapwithnonemptyclosed

5 Derived cones to reachable sets of a class of second-order differential inclusions 5 values, which define the following Cauchy problem associated to a second-order differential inclusion 2.1) x Ax + F t, x), x) = x, x ) = x 1. A continuous mapping x ) CI,X)iscalledamild solution of problem 2.1) if there exists a Bochner) integrable function f ) L 1 I,X) such that 2.2) ft) F t, xt)) a.e. I), 2.3) xt) =Ct)x + St)x 1 + t St u)fu)du t I, i.e., f ) is a Bochner) integrable selection of the set-valued map F,x )) and x ) is the mild solution of the Cauchy problem 2.4) x = Ax + ft), x) = x, x ) = x 1. We shall call x ),f )) a trajectory-selection pair of 2.1) if f ) verifies 2.2) and x ) is a mild solution of 2.4). As an example, consider the following Cauchy problem associated to a nonlinear wave equation 2.5) 2 z t 2 t, x) 2 z t, x) Gzt, x)) x2 zt, ) = zt, π) = in,t), in,t),π), z z,x)=z x), t,x)=z 1x) a.e. x,π), where G ) :R PR) is a set-valued map, z ) H 1,π), z 1 ) L 2,π). Problem 2.5) may be rewritten as an ordinary differential inclusion of the form 2.1) in X = L 2,π), where A : DA) X X is defined by Az = d2 z, DA) dt 2 =H1,π) H2,π)andF ) :X PX) isgivenby F z) :=SelGz )) for each z H 1,π). Here Sel Gz )) is the set of all f L 2,π) satisfying fx) Gzx)) a.e. for x,π). We recall that Au = n=1 n2 u, u n u n, u DA), where u n t) = 2 π ) 1 2 sinnt), n =1, 2,... and A is the infinitesimal generator of the cosine family {Ct); t R} defined by Ct)u = n=1 cos nt u, u n u n,u X. The sine family associated is given by St)u = n=1 1 n sin nt u, u n u n,u X. Hypothesis 2.3. i) F, ) :I X PX) has nonempty closed values and is LI) BX) measurable. ii) There exists L ) L 1 I,R + ) such that, for any t I,Ft, ) is Lt)- Lipschitz in the sense that d H F t, x 1 ),Ft, x 2 )) Lt) x 1 x 2 x 1,x 2 X.

6 Aurelian Cernea 6 The main tool in characterizing derived cones to reachable sets of semilinear differential inclusions is a certain version of Filippov s theorem for differential inclusion 2.1). Hypothesis 2.4. Let S be a separable metric space, X,X 1 X are closed sets, a ) :S X, a 1 ) :S X 1 and c ) :S, ) are given continuous mappings. The continuous mappings g ) :S L 1 I,X), y ) :S CI,X) are given such that ys)) t) =Ays)t)+gs)t), ys)) X, ys)) ) X 1. There exists a continuous function p ) :S L 1 I,R + ) such that dgs)t),ft, ys)t))) ps)t) a.e.i), s S. Theorem 2.5 [8]). Assume that Hypotheses 2.3 and 2.4 are satisfied. Then there exist M>and the continuous functions x ) :S L 1 I,X), h ) :S CI,X) such that for any s S xs) ),hs) )) is a trajectoryselection of 1.1) satisfying for any t, s) I S 2.6) xs)) = a s), xs)) ) = a 1 s), [ 2.7) xs)t) ys)t) M cs)+ a s) ys)) + t ] + a 1 s) ys)) ) + ps)u)du. 3. THE MAIN RESULTS Our object of study is the reachable set of 1.1) defined by R F T,X,X 1 ):={xt ); x ) is a mild solution of 1.1)}. We consider a certain variational second-order differential inclusion and we shall prove that the reachable set of this variational inclusion from derived cones C X to X and C 1 X to X 1 at time T is a derived cone to the reachable set R F T,X,X 1 ). Throughout in this section we assume Hypothesis 3.1. i) Hypothesis 2.3 is satisfied and X,X 1 X are closed sets. ii) z ),f )) CI,X) L 1 I,X) is a trajectory-selection pair of 1.1) and a family P t, ) :X PX), t I of convex processes satisfying the condition 3.1) P t, u) Q ft) F t, )zt); u) u domp t, )), a.e. t I

7 Derived cones to reachable sets of a class of second-order differential inclusions 7 is assumed to be given and defines the variational inclusion 3.2) v Av + P t, v). Remark 3.2. We note that for any set-valued map F, ), one may find an infinite number of families of convex process P t, ), t I, satisfying condition 3.1); in fact any family of closed convex subcones of the quasitangent cones, P t) Q zt),ft)) GraphF t, )), defines the family of closed convex process P t, u) ={v X; u, v) P t)}, u,v X, t I that satisfy condition 3.1). One is tempted, of course, to take as an intrinsic family of such closed convex process, for example Clarke s convex-valued directional derivatives C ft) F t, )zt); ). We recall e.g. [1]) that, since F t, ) is assumed to be Lipschitz a.e. on I, the quasitangent directional derivative is given by { 1 } 3.3) Q ft) F t, )zt); u))= w X; lim θ + θ dft)+ θw, Ft, zt)+ θu))=. Weareablenowtoprovethemainresultofthispaper. Theorem 3.3. Assume that Hypothesis 3.1 is satisfied, let C X be a derived cone to X at z) and C 1 X be a derived cone to X 1 at z ). Then the reachable set R P T,C,C 1 ) of 3.2) is a derived cone to R F T,X,X 1 ) at zt ). Proof. In view of Definition 2.1, let {v 1,...,v m } R P T,C,C 1 ). Hence there exist the trajectory-selection pairs u 1 ),g 1 )),...,u m ), g m )) of the variational inclusion 3.2) such that 3.4) u j T )=v j, u j ) C, u j) C 1, j =1, 2,...,m. Since C X is a derived cone to X at z) and C 1 X is a derived cone to X 1 at z ), there exist the continuous mappings a : S =[,θ ] m X, a 1 : S X 1 such that 3.5) a ) = z), Da )s = a 1 ) = z ), Da 1 )s = s j u j ) s R m +, s j u j) s R m +.

8 Aurelian Cernea 8 Further on, for any s =s 1,...,s m ) S and t I we denote 3.6) ys)t) =zt)+ s j u j t), gs)t) =ft)+ s j g j t), ps)t) =dgs)t),ft, ys)t))) and prove that y ), p ) satisfy the hypothesis of Theorem 2.5. Using the lipschitzianity of F t,, ) we have that for any s S, the measurable function ps) ) in 3.6) it is also integrable. ps)t) =dgs)t),ft, ys)t))) s j g j t) +d H F t, zt)),ft, ys)t))) s j g j t) + Lt) s j u j t). Moreover, the mapping s ps) ) L 1 I,R + ) is continuous in fact Lipschitzian) since for any s, s S onemaywritesuccessively ps) ) ps ) ) 1 = T T ps)t) ps )t) dt [ gs)t) gs )t) dt +d H F t, ys)t)),ft, ys )t)))) ] dt m T s s [ gj t) + Lt) u j t) ] ) dt. Let us define S 1 := S \{,...,)} and c ) :S 1, ), cs) := s 2. It follows from Theorem 2.5 the existence of a continuous function x ) :S 1 CI,X) such that for any s S 1, xs) ) is a mild solution of 1.1) with the properties 2.6) 2.7). For s = we define x)t) =y)t) =zt) t I. Obviously, x ) : S CI,X) is also continuous. Finally, we define the function a ) :S R F T,X,X 1 )by as) =xs)t ) s S. Obviously, a ) is continuous on S and satisfies a) = zt ).

9 Derived cones to reachable sets of a class of second-order differential inclusions 9 To end the proof we need to show that a ) is differentiable at s = S and its derivative is given by Da)s) = s j v j s R m + which is equivalent with the fact that 1 as) m ) 3.7) lim a) s j v j =. s s From 2.7) we obtain 1 1 as) a) s j v j s s xs)t ) ys)t ) M s + M a s) s z) s j u j ) + M a 1 s) z ) s s j u j) + M T ps)u) s du and therefore in view of 3.5), relation 3.7) is implied by the following property of the mapping p ) in 3.6) ps)t) 3.8) lim = a.e. I). s s In order to prove the last property we note since P t, ) is a convex process for any s S one has s j s g jt) P t, ) s j s u jt) Q ft) F t, ) zt); Hence by 3.3) we obtain 1 3.9) lim h + h d s j ft)+h s g jt),f t, zt)+h ) s j s u jt) a.e. I). )) s j s u jt) =. In order to prove that 3.9) implies 3.8) we consider the compact metric space S+ m 1 = {σ R m + ; σ =1} and the real function φ t, ) :,θ ] S+ m 1 R + defined by 3.1) φ t h, σ) = 1 )) ft)+h h d σ j g j t),f t, zt)+h σ j u j t), where σ =σ 1,...,σ m ) and which according to 3.9) has the property 3.11) lim θ + φ tθ, σ) = σ S m 1 + a.e. I).

1 Aurelian Cernea 1 Using the fact that φ t θ, ) is Lipschitzian and the fact that S+ m 1 is a compact metric space, from 3.11) it follows easily e.g. Proposition 4.4 in [13]) that lim θ + max σ S m 1 + φ t θ, σ) = which implies that ) lim φ s t s, = a.e. I) s s and the proof is complete. An application of Theorem 3.3 concerns local controllability of the secondorder differential inclusion in 1.1) along a reference trajectory, z ) attimet, in the sense that zt ) intr F T,X,X 1 )). Theorem 3.4. Let X = R n, z ),F, ) and P, ) satisfy Hypothesis 3.1, let C X be a derived cone to X at z) and C 1 X be a derived cone to X 1 at z ). If the variational second-order differential inclusion in 3.2) is controllable at T in the sense that R P T,C,C 1 )=R n, then the differential inclusion 1.1) is locally controllable along z ) at time T. Proof. The proof is a straightforward application of Lemma 2.2 and of Theorem 3.3. REFERENCES [1] J.P. Aubin and H. Frankowska, Set-valued Analysis. Birkhäuser, Berlin, 199. [2] M. Benchohra, On second-order differential inclusions in Banach spaces. NewZealand J. Math. 3 21), 5 13. [3] M. Benchohra and S.K. Ntouyas, Existence of mild solutions of second-order initial value problems for differential inclusions with nonlocal conditions. Atti.Sem.Mat.Fis. Univ. Modena 49 21), 351 361. [4] A. Cernea, Local controllability of hyperbolic differential inclusions via derived cones. Rev. Roumaine Math. Pures Appl. 47 22), 21 31. [5] A. Cernea, Derived cones to reachable sets of differential-difference inclusions. Nonlinear Anal. Forum 11 26), 1 13. [6] A. Cernea, Derived cones to reachable sets of discrete inclusions. Nonlinear Studies 14 27), 177 187. [7] A. Cernea, On a second-order differential inclusion. Atti.Sem.Mat.Fis.Univ.Modena 55 27), 3 12. [8] A. Cernea, Continuous version of Filippov s theorem for a second-order differential inclusion. An. Univ. Bucureşti Mat. 57 28) 3 12. [9] A. Cernea, On the solution set of some classes of nonconvex nonclosed differential inclusions. Portugaliae Math. 65 28), 485 496.

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