STABILITY ANALYSIS OF SOLUTIONS TO AN OPTIMAL CONTROL PROBLEM ASSOCIATED WITH A GOURSAT-DARBOUX PROBLEM

Transcription

1 Int. J. Appl. Math. Comput. Sci., 3, Vol. 13, No. 1, 9 44 STABILITY ANALYSIS OF SOLUTIONS TO AN OTIMAL CONTROL ROBLEM ASSOCIATED WITH A GOURSAT-DARBOUX ROBLEM DARIUSZ IDCZAK, MAREK MAJEWSKI STANISŁAW WALCZAK University of Łódź, Faculty of Mathematics ul. S. Banacha, 9 38 Łódź, oland {idcza, marmaj, stawal}@imul.math.uni.lodz.pl In the present paper, some results concerning the continuous dependence of optimal solutions and optimal values on data for an optimal control problem associated with a Goursat-Darboux problem and an integral cost functional are derived. Keywords: Goursat-Darboux problem, optimal control, continuous dependence 1. Introduction The question of the continuous dependence of solutions to an optimal control problem on data the so-called stability analysis of solutions or well-posedness) is very important from the point of view of practical applications of theory. Indeed, if the problem is related to a physical phenomenon, its data can be considered only an arbitrarily close approximation to exact values. Consequently, if the solution does not depend continuously on the data, it is not actually determined. Example 1. Let us consider the one-dimensional system with a scalar parameter ω [1/, 1] ẋ 1 t) x t), ẋ t) ut) ω x 1 t), u t) [, 1] with boundary conditions x 1 ), x 1 π), x ) and x π) are free, and the cost functional π J ω x, u) x 1 t) x 1 t) 1 3 π ) dt inf. In a way analogous to Idcza, 1998) one can show that for any parameter ω [1/, 1] the above optimal control problem possesses an optimal solution x ω, u ω). Supported by grant 7 T11A 4 1 of the State Committee for Scientific Research, oland. One can show that for any ω [1/, 1) J ω x ω, u ω) π π. If ω 1, then the control system has a solution x ω not unique) only for u. Moreover, J ω x ω, u ω) So, we see that as ω 1, the optimal value has a jump, i.e. it is not continuous with respect to ω at the point ω 1. In this case, we say that the optimal control problem is ill posed. The stability analysis of solutions to finitedimensional mathematical programming problems was investigated, e.g. in Ban et al., 1983; Fiacco, 1981; Levitin, 1975; Robinson, 1974). A survey of the results for the case of abstract Banach spaces is given in Malanowsi, 1993). We study the continuous dependence of solutions and optimal values on data for the optimal control problem associated with the Goursat-Darboux problem w x y g x, y, w, w x, w ) y, u, x, y) [, 1] [, 1] a.e., wx, ) ϕx), w, y) ψy), x, y [, 1], 1 1 NH) Iw, u) Gx, y, w, u) dx dy min, u U M { u L ); ux, y) M, x, y) a.e. } in linear and nonlinear cases).

2 3 Systems of this type in control theory they are called two-dimensional D) continuous Fornasini-Marchesini systems) have been investigated by many authors see, e.g. Bergmann et al., 1989; ulvirenti and Santagati, 1975; Surryanarayama, 1973)). They can be applied to describe the absorption gas phenomenon Idcza and Walcza, 1994; Tihonov and Samarsi, 1958). Our aim is to obtain stability results analogous to those obtained in Walcza, 1) for an ordinary problem. The main tools used in the study of the stability question in the case of abstract Banach spaces are: the implicit function theorem for generalized equations Robinson, 198), the open mapping theorem for setvalued maps Robinson, 1976), and composite optimization Ioffe, 1994). Our proofs mae no appeal to the above approaches. First, we consider roblem NH) in the case when the system is linear, autonomous, and the set M does not vary. The approach used here is based on the Cauchy formula for a solution to a linear autonomous system. Since, in the nonlinear case, we have no formula for solving system NH), in this case we use a different method, based on the Gronwall lemma for functions of two variables and the continuity of the mapping M U M in the Hausdorff sense). Let us point out the fact that this method cannot be applied in the linear case.. Space of Solutions to the Goursat Darboux roblem By a function of an interval Łojasiewicz, 1988) we mean a mapping F defined on the set of all closed intervals [x 1, x ] [y 1, y ] contained in, with values in R. We say that F is additive if F Q R) F Q) F R) for any closed intervals Q, R such that Q R is an interval contained in and IntQ IntR. Let a function z : R of two variables be given. The function F z of an interval given by F z [x1, x ] [y 1, y ] ) z x, y ) z x 1, y ) z x, y 1 ) zx 1, y 1 ) for [x 1, x ] [y 1, y ] is called the function of an interval associated with z. We say that a function z : R of two variables is absolutely continuous Walcza, 1987) if z, ), z, ) are absolutely continuous functions on [, 1] and F z is an absolutely continuous function of an interval. It can be shown Walcza, 1987) that z : R is absolutely continuous if and only if there exist functions D. Idcza et al. l L 1, R), l 1, l L 1 [, 1], R) and a constant c R such that z x, y) for x, y). y l s, t) ds dt x l 1 s) ds l t) dt c 1) The above integral formula implies that the partial derivatives z z x, y), x exist a.e. on and x, y), y z x, y) x y z y x, y) l x, t) dt l 1 x), x z x, y) y x z x, y) l x, y) x y for x, y) a.e. Idcza, 199). l s, y) ds l y), Moreover, it is easy to see that a function z : R is absolutely continuous and satisfies the conditions z, y) for y [, 1], z x, ) for x [, 1], if and only if there exists a function l L 1, R) such that for x, y). z x, y) l s, t) ds dt ) By AC, R) we denote the set of all functions z : R, each of them having the integral representation 1) with functions l L, R), l 1, l L [, 1], R). By AC, R) we denote the set of all functions z : R each of them having the integral representation ) with function l L, R). By AC, R n ) AC, R n )) we denote the set of all vector-valued functions z z 1, z,..., z n ) : R n such that each coordinate function z i : R belongs to AC, R) AC, R)).

3 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem 31 It is easy to see that AC, R n ) with the scalar product z, w AC,R n ) l s, t), s, t) ds dt, R n where z x, y) w x, y) l s, t) ds dt, s, t) ds dt and AC, R n ) with the scalar product z, w AC,R n ) where zx, y) wx, y) 1 1 y y l s, t), s, t) R n ds dt l 1 s), 1 s) R n ds l t), t) R n dt c, d R n, ls, t) ds dt l t) dt c, s, t) ds dt t) dt d x x l 1 s) ds 1 s) ds are Hilbert spaces. In much the same way as in Idcza, 1996) it can be proved that if z n z wealy in AC, R n ), then z n z uniformly on. In the sequel, we denote by AC [, 1], R n ) the standard space of all absolutely continuous functions of one variable ϕ : [, 1] R n such that ϕ L [, 1], R n ). The scalar product in AC [, 1], R n ) is given by 1 ϕ, ψ AC [,1],R n ) l s), s) R n ds c, d R n, where ϕ x) ψ x) 3. Linear System x x l s) ds c, s) ds d roblem Formulation and Basic Assumptions Consider the family of optimal control problems v x y x, y) v A 1 x x, y) v A x, y) y A v x, y) B u x, y), x, y) a.e., v x, ) ϕ x), v, y) ψ y) L ) for all x, y [, 1], J v, u) G x, y, v x, y), u x, y)) dx dy, where A 1, A, A R n n, B R n m, ϕ, ψ AC [, 1], R n ) and ϕ ) ψ ) c for, 1,,.... roblem L ) is considered in the space AC, R n ) of solutions v and in the set U M {u L, R m ) : ux, y) M a.e.} of controls u. The set M is a convex and compact subset of R m. It is easy to see that, using the substitution z x, y) v x, y) ϕ x) ψ y) c, we get the equivalent problem z x y x, y) z A 1 x x, y) z A x, y) y A z x, y) B u x, y) b x, y), LH z x, ) z, y) for all x, y), ) J z, u) F x, y, z x, y), u x, y)) dx dy, where b x, y) A d 1 dx ϕ x) A d dx ψ y) A ϕ x) A ψ x) A c, F x, y, z, u) G x, y, zϕ x)ψ y) c, u ). roblem LH ) will be considered in the space AC, R n ) of solutions z and in the set U M of controls u.

4 3 For simplicity, we prove our results for roblem LH ). However, roblems LH ) and L ) are equivalent, and therefore all results which are to be proved can be used for L ). For the system z x y x, y) z A 1 x x, y) z A x, y) y A z x, y) B u x, y) b x, y), x, y) a.e., z x, ) z, y) for all x, y [, 1], the following theorem holds Bergmann et al., 1989): Theorem 1. For any u L, R m ) the system 3) possesses a unique solution z AC, R n ) given by the formula z x, y) R s, t, x, y) B u s, t) b s, t) ) ds dt, 3) 4) where the function R : R n n called the Riemann function) has the form R s, t, x, y) i j s x) i t y) j i! j! T i,j, and the sequence Ti,j is defined by the recurrence formulae T i,j T i,j 1 A 1 T i 1,j A T i 1,j 1 A, T, I, T i,j for i 1 or j 1, 5) for, 1,,.... We shall mae the following assumptions: L) b b in L,R n ), L1) the function x, y) F x, y, z, u) R is measurable for z, u) R n R m, and, 1,,..., L) the function R n R m z, u) F x, y, z, u) R is continuous for x, y) a.e.,, 1,,..., L3) the function R m u F x, y, z, u) R is convex for x, y) a.e., z R n and, 1,,..., D. Idcza et al. L4) for any bounded set B R n, there exists a function γ B L 1, R ) such that F x, y, z, u) γ B x, y) u for x, y) a.e., z B, u R m and, 1,,..., L5) the sequences of matrices A ) N, A 1) N, A ) N tend to matrices A, A 1, A, respectively, in the norm of R n n, and B ) N tends to B in the norm of R n m by the norm of a matrix A we mean the value A a i,j ) 1 ). i,j For, 1,,..., set Z { z AC, R n ) : and there exists u U M such that z is the solution of 3) corresp. to u} m inf J z, u) with respect to z, u) such that z is the solution of 3) corresponding to u U M. In a standard way one can prove the following result: Theorem. Assume that L1) L4) hold. Then, for any, 1,,..., there exists an optimal solution of roblem LH ), i.e. for any, 1,,... there exist control u U M and the trajectory z Z corresponding to u, such that J z, u ) m. Write A { z, u ) AC, R n ) U M : z satisfies 3) with u and J z, u ) m }. 6) This set will be referred to as the set of optimal solutions, or the optimal set. Lemma 1. Let Then c sup max{ A, A 1, A, B, 1}. {,1,... } 1. for any, 1,,..., for x, y, s, t), R x, y, s, t) e 3c

5 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem 33. if ε > and K are such that, for any > K, A A ε 3, A 1 A 1 ε 3, A A ε 3, then, for > K, we have T i,j Ti,j ε3 ij1 c ij for i, j {, 1,,... } and, consequently, R x, y, s, t) R x, y, s, t) 3εe 3c for any x, y, s, t) i.e. R R uniformly on as ). The proof of this lemma using the induction argument) has only a technical character and is very arduous. Let us recall that the wea upper limit of a sequence of the sets V X X is a Banach space) is defined as the set of all cluster points with respect to the wea topology) of sequences v ) where v V for 1,, 3,.... We denote this set as wlimsup V Aubin and Franowsa, 199). Theorem 3. If 1. roblems LH ) satisfy the conditions L) L5),. the sequence of cost functionals J x, u) tends to J x, u) uniformly on B U M for any bounded set B AC, R n ), then a) there exists a ball B, ρ) AC, R n ) such that Z B, ρ) for, 1,,..., i.e. there exists ρ > such that, for any z Z, we have z AC,R n ) ρ, b) the sequence of optimal values m tends to an optimal value m, c) the wea upper limit of the optimal sets A AC, R n ) L, R m ) is a non-empty set, and wlimsup A A. If the set A is a singleton, i.e. A {z, u )} for, 1,,..., then z tends to z wealy in AC, R n ) and u tends to u wealy in L, R m ). roof. a) Let z AC, R n ) be the solution of 3) corresponding to u. From 4) we have z x y x, y) B u x, y) b x, y) x x R s, y, x, y) B u s, y) b s, y) ) ds y y R x, t, x, y) B u x, t) b x, t) ) dt x y R s, t, x, y) B u s, t) b s, t) ) ds dt for x, y). Since u x, y) M, which is bounded, B B and R are analytic, from L) and Lemma 1 we get that there exists a constant ρ > such that z AC,R n ) for, 1,,.... z x, y) x y ) 1 dx dy ρ b) Let z, u ) A for 1,, 3,.... Then we have m J z, u ) for 1,, 3,..., where z is the trajectory of 3) with, corresponding to u for 1,,.... Let ε >. By L) and L5), there exists a K such that, for any > K, A A ε 3, A 1 A 1 ε 3, A A ε 3, B B < ε, b x, y) b x, y) dx dy < ε. 7) By direct calculations, from Lemma 1 and 7) we obtain, for > K and x, y), z x, y) z x, y) R s, t, x, y) R s, t, x, y) B u s, t) b s, t) ds dt

6 34 R s, t, x, y) B B u s, t) ds dt R s, t, x, y) b s, t) b s, t) ds dt ε c, where c >. This means that sup x,y) z x, y) z x, y) is arbitrarily small for a sufficiently large ). From the Scorza-Dragoni Theorem Eeland and Temam, 1976) applied to the function F B M where B is the ball with radius ρ described in a)) we have that for any η > there exists a compact set η such that µ \ η ) η and F η B M is uniformly continuous. Thus for a sufficiently large we have J z, u ) J ) z, u F x, y, z x, y), u x, y) ) η F x, y, z x, y), u x, y) ) F x, y, z x, y), u x, y) ) \ h F x, y, z x, y), u x, y) ) µ η ) η ηĉ ε where ĉ > and ε is arbitrarily small. Thus, for any ε >, m J z, u ) J z, u ) ε 8) for a sufficiently large. From Assumption and a) we have J z, u ) J ) z, u < ε 9) for a sufficiently large. Consequently, by 8) and 9), m J z, u ) ε m ε for a sufficiently large. Similarly, we can prove that, for sufficiently large, m m ε. We have thus proved that m m as. c) Let z, u ) be an optimal process for LH ), i.e. z, u ) A for, 1,,.... Since u x, y) M and M is compact, u ) N is bounded. Since L, R m ) is reflexive, the sequence u ) N is compact in the wea topology of the space L, R m ). Without loss of generality, we can assume that u ū U M in D. Idcza et al. the wea topology. From the formula of solution 4) we have that, for any x, y), z x, y) R s, t, x, y) B u s, t) b s, t) ) ds dt R s, t, x, y) R s, t, x, y) ) B u s, t) b s, t) ) ds dt R s, t, x, y) B B ) u s, t) b s, t) b s, t) ) ds dt R s, t, x, y) B u s, t) b s, t) ) ds dt. By virtue of Lemma 1 we have R R. By L), L5) and from the boundedness of M the first and the second integral tend to zero. By the wea convergence of u, the last integral tends to R s, t, x, y) B s, t) u s, t) b s, t) ) ds dt. In this way we have proved that z tends pointwisely to some z AC, R n ) which is the solution to LH ) with, corresponding to ũ. Further, we prove that z, ũ ) is an optimal process for LH ) with. Suppose that it is not true. Let z, u ) be an optimal process for LH ) with. Let J z, ũ ) J z, u ) α >. 1) Then we have m m J z, u ) J z, u ) [ J z, u ) J z, u )] [ J z, u ) J z, ũ )] α. By b), m m. From Assumption ) and a) we get that the first component tends to zero as. Moreover, lim J z, u ) J z, ũ ) by L) L4). In this way we have a contradiction with 1) and the proof is complete.

7 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem Main Results for a Linear System Based on Theorem 3, we obtain the following sufficient conditions for the stability of a two-dimensional optimal control system: Corollary 1. Suppose that, for any, 1,..., roblem LH ) satisfies Assumptions L) L5) and, for any bounded set B R n, there exists a sequence of functions γ B L1, R ) such that F x, y, z, u) F x, y, z, u) γ B x, y) for x, y) a.e., z, u) B M and, 1,,.... Moreover, we assume that γ B in L1, R ). Then a) the sequence of optimal values m tends to an optimal value m as, b) wlimsup A A and wlimsup A. roof. Let B be any bounded set in the space AC, R n ). It is easy to see that if z B, then z x, y) B R n for any x, y), where B is bounded. By assumption, we have that, for any z, u) B U M, J z, u) J z, u) F x, y, z x, y), u x, y)) F x, y, z x, y), u x, y)) γb x, y) dx dy. Since γ B in L1, R ), the sequence of cost functionals J converges uniformly to J on B U M for any bounded set B AC, R n ). In this way, the assumptions of Theorem 3 are fulfilled and, by this theorem, a) and b) are true. Corollary. If 1. we have F x, y, z, u) G 1 x, y, ω x, y), z ) G x, y, ω x, y), z ), u where ω ) L p, R s ), p 1 and functions R s R n x, y, ω, z) G 1 x, y, ω, z) R, R s R n x, y, ω, z) G x, y, ω, z) R m are measurable with respect to x, y) and continuous with respect to ω, z),. ω ω in the norm topology of L p, R s ) as, 3. for any bounded set B R n, there exists C > such that G i x, y, ω, z) C 1 ω p ) for x, y) a.e., ω R s, z B, 4. roblems LH ) satisfy Assumptions L) L5), then the conditions a) and b) of Corollary 1 hold. roof. We shall prove that the sequence of cost functionals J z, u) tends to J z, u) uniformly on any set B UM where B AC, R n ) is bounded. Suppose that this is not true. Then there exist some bounded set B AC, R n ), ε > and some sequence z i, u i ) i N, such that z i B, u i U M and J i z i, u i) J z i, u i) ε 11) for i 1,, 3,.... By the reflexivity of AC, R n ), we may assume extracting, if necessary, a subsequence) that z i tends to some z uniformly on as i. We have J i z i, u i) J z i, u i) G1 x, y, ω i x, y), z i x, y) ) G 1 x, y, ω x, y), z x, y) ) dx dy G1 x, y, ω x, y), z x, y) ) G 1 x, y, ω x, y), z i x, y) ) dx dy G x, y, ω i x, y), z i x, y) ) G x, y, ω x, y), z x, y) ) u i x, y) dx dy G x, y, ω x, y), z x, y) ) G x, y, ω x, y), z i x, y) ) u i x, y) dx dy. Since u i U M, it is commonly bounded. By Krasnoselsii s theorem on the continuity of the Nemytsii operator, the right-hand side of the above inequality tends to zero as i. This contradicts 11), and we have thus proved that the sequence of cost functionals J z, u) tends to J z, u) uniformly on any set B UM, where B AC, R n ) is bounded. Applying Theorem 3), we complete the proof.

8 36 Next, let us consider a mixed case, i.e. when the cost functional is of the form J z, u) G1 x, y, z x, y)), ω x, y) G x, y, v x, y), z x, y)), u x, y) dx dy, 1) where G 1 : R n R s, G : R r R n R m,, 1,,.... Corollary 3. If 1. the cost functional is of the form 1),. the function G 1 is measurable with respect to x, y) and continuous with respect to z analogously G ), 3. for any bounded set B R n, there exist α ) L p, R ) and C >, such that G 1 x, y, z) α x, y), G x, y, v, z) 1 v p ) for x, y) a.e., z B and v R r, 4. ω tends to ω in the wea topology of L q, R s ) when q [1, ), or wealy- when q ; v tends to v in the norm topology of L p, R s ), 5. the optimal control problems LH ) satisfy Assumptions L) L5), then the optimal values and the sets of optimal processes satisfy the conditions a) and b) of Corollary 1. Remar 1. The obtained results remain true for a family of problems L ), whereas the following additional assumption concerning the functions ϕ, ψ is satisfied: ϕ ϕ, ψ ψ in AC [, 1], R n ). Example. Consider a two-dimensional continuous optimal control system with variable parameters z xy x, y) A 1z x x, y) 1 A) zy x, y) 1 B ) u x, y), 13) z x, ) ϕ x), z, y) ψ y), u x, y) [, 1], J z, u) [ x ) z ω 1 x, y) φ 1 x, y, z x, y)) ω x, y) φ x, y, u x, y)) x) u x, y) 4xy ]dx dy min, D. Idcza et al. where ϕ, ψ AC [, 1], R), φ 1 is continuous and φ is continuous and convex with respect to u [, 1], u ) L, [, 1]), z AC, R), ω 1 ), ω ) L 1, [ 1, 1]). By Theorem, the problem 13), 14) possesses at least one optimal solution but, in general, it is not easy to find an optimal process for this system. Suppose that A 1, A, B in R; ϕ, ψ in AC [, 1], R), ω 1, ω in L 1, R) as. In the limit case, we obtain the problem J z, u) z xy x, y) z y x, y) u x, y), z, y) z, y), [ x ) z x, y) 14) x) u x, y)4xy ]dx dy. 15) By Theorem, the above problem possesses an optimal process and, applying the extremum principle, we are able to find effectively an optimal solution z, u ) and an optimal value m. In fact, the Lagrange function for the system 14), 15) is of the form L z, u) [ x ) z x, y) 1 1 x) u x, y) 4xy 4 v x, y) z xy x, y) z y x, y) u x, y) where v L, R). ] dx dy 16) The extremum principle implies that L z z, u ) h for any h AC, R n ), and for any admissible control u. L z, u ) L z, u) 17) Taing account of 16) and integrating by parts, we get L z z, u ) h x ) h x, y) v x, y) h xy x, y) h y x, y)) dx dy [ x x y x ) dx dy v x, y) ] v x, y) dx h xy x, y) dx dy

9 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem 37 for any h AC, R n ). Thus v x, y) 1 x v x, y) dx1 y) 3 x 1 ) x. The above equation is of the Volterra type, and therefore there exists a unique solution v of this equation. By direct calculation, it is easy to chec that v x, y) 1 x) 1 y). The minimum condition 17) taes the form 1 x) y 3 ) u x, y) dx dy 4 1 x) y 3 ) u x, y) dx dy 4 for all u x, y) [, 1]. This implies 1 for x [, 1], y [ ], 3 u 4 x, y) for x [, 1], y [ 18) 3 4, 1] and that, by 14) and 15), 1 e x ) y for x [, 1], y [ ], 3 z 4 x, y) 1 e x ) 3 4 for x [, 1], y [ 3 4, 1] 19) and m J z, u ) Applying Theorem to our example, we see that, for any 1,, 3,..., there exists at least one optimal process z, u ) for the system 13), 14), and the sequence u ) N tends to u wealy in L, z ) N tends to z wealy in AC, R n ), where u and z are defined by 18) and 19), respectively. Moreover, the sequence m ) of the optimal values for the systems 13), 14) tends to m 55/64 and the sequence ) z of N the optimal trajectories tends to z uniformly on. In this way, we deduce that, in general, it is difficult to find an optimal solution for 13), 14), but the process z, u ) given by 18), 19) and the optimal value m 55/65 are a good approximation for z, z ) and m with a sufficiently large. 4. Nonlinear System 4.1. reliminaries In this part we give a definition of a Hausdorff metric and prove some generalization of the Gronwall lemma regarding the case of functions of two variables. Let X, ρ) be a metric space. We define a distance from a point x X to a bounded set A X as distx, A) inf{ρx, a); a A}. The Hausdorff distance ρ H A, B) between the bounded sets A, B X is defined as ρ H A, B) inf{ε > ; A NB, ε), B NA, ε)} where, for a given ε > and a bounded set C X, NC, ε) {x X; distx, C) ε}. The function ρ H restricted to the set Z Z, where Z is the family of all closed bounded subsets of X, is a metric in Z Kisielewicz, 1991). It is called the Hausdorff metric. Lemma. If, c >, υ : R is continuous and then υx, y) roof. Write ux, y) υs, t) ds dt c, x, y), υx, y) e xy c, x, y). υs, t) ds dt c, x, y), wx, y) e xy ux, y), x, y). From the continuity of υ it follows that u possesses the partial derivatives ux, y)/ x and ux, y)/ y everywhere on. Moreover, u y x, y) υx, t) dt x y x t ) υs, τ) ds dτ c dt y τ)υs, τ) ds dτ cy

10 38 for x, y). Consequently, w x x, y) ye xy ux, y) e ye xy ux, y) e xy y e xy xy u υs, τ) ds dτ x, y) x τυs, τ) ds dτ e xy cy ye xy ux, y) ye xy ux, y) e xy for x, y) and, analogously, Consequently, for x, y). This implies τυs, τ) ds dτ w x, y), x, y). y wx, y) w, ) c υx, y) ux, y) e xy wx, y) e xy c for x, y). 4.. Main Result In this part we consider the family of homogeneous problems z x y f x, y, z, z x, z ) y, u, x, y) a.e., zx, ), z, y), x, y [, 1], J z, u) 1 1 F NH ) x, y, z, u) dx dy min, u U { u L, R m ); ux, y) M, x, y) a.e.},, 1,.... roblem NH ) will be referred to as the limit problem. In the sequel, we shall assume that the functions f : R n R n R n M R n, F : R n M R, D. Idcza et al. where M M, and the sets M R m,, 1,..., satisfy the following conditions: N1) the sets M are compact and M M in R m with respect to the Hausdorff metric; N) the functions f are measurable in x, y) a.e., continuous in u M and there exists a constant L > such that f x, y, z, z x, z y, u) f x, y, w, w x, w y, u) L z w z x w x z y w y ) for x, y) a.e., z, z x, z y, w, w x, w y R n, u M,, 1,... ; N3) there exist constants a, b > such that f x, y, z, z x, z y, u) a z b for x, y) a.e., z, z x, z y R n, u M,, 1,... ; N4) for any bounded set A R n R n R n, there exists a sequence ) ϕ A N L, R ) such that ϕ A in L, R ) and f x, y, z, z x, z y, u) f x, y, z, z x, z y, u) for x, y) a.e., 1,,... ; N5) the function f is of the type f x, y, z, z x, z y, u) ϕ Ax, y) α x, y, z, z x, z y ) β x, y, z, z x, z y )u, where β is measurable in x, y), continuous in z, z x, z y ) R n R n R n and, for any bounded set A R n R n R n, there exists a function γ A L, R ) such that β x, y, z, z x, z y ) γ A x, y) for x, y) a.e., z, z x, z y ) A; N6) the functions F are measurable in x, y), continuous in z, u) R n R m and, for each bounded set B R n, there exists a function ν B L 1, R ) such that F x, y, z, u) νb x, y) for x, y) a.e., z B, u M,, 1,... ;

11 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem 39 N7) for any ) bounded set B R n, there exists a sequence ψ B N L1, R ) such that ψb in L 1, R ) and F x, y, z, u) F x, y, z, u) ψbx, y) for x, y) a.e., z B, u M, 1,,.... Remar. From Assumption N1) it follows that the sets M,, 1,..., are commonly bounded in R m. Remar 3. From Assumption N) it follows that the functions f are continuous in z, z x, z y, u) R n R n R n M. In much the same way as in Idcza et al, 1994), one can show that there exist l N and α, 1) such that each operator F u : L, R n ) g f x, y, y g, g, x ) g, ux, y) L, R n ), where u U,, 1,..., is contracting with the constant α, 1) with respect to the Bieleci norm in L, R n ) given by g l 1 1 ) 1 e lxy) gx, y) dx dy, g L, R n ). Consequently, Fu possesses a unique fixed point gu L, R n ). This means that the system z x y f x, y, z, z x, z ) y, u has a unique solution z u in the space AC, R n ). This solution is given by and z ux, y) z u x y x, y) g ux, y), g us, t) ds dt, x, y), x, y) a.e. Let us recall that the wea convergence in AC, R n ) implies the uniform convergence. Using the standard arguments, one can also easily show that if z n z in AC, R n ), then z n / x z / x in L, R n ) as n. In the proof of the main theorem, we shall use the following three lemmas: Lemma 3. There exists a constant r such that zu x, y) x y, zu x, y) x, zu x, y) y r for x, y) a.e. and u U,, 1,.... roof. From Assumption N3) we have zu x, y) x y f x, y, zux, y), z u x x, y), z u x, y), ux, y) y a z ux, y) b, x, y) a.e. for u U,, 1,.... Hence z u x, y) x a y z u x y s, t) ds dt z u s, t) ds dt b, x, y) for u U,, 1,.... Applying the previous lemma with υx, y) z ux, y), a, c b, we obtain z ux, y) e a b, x, y), for u U,, 1,.... Thus zu x, y) x y aea b b, x, y) a.e., for u U,, 1,.... The remaining part of the assertion follows from the fact that zu y zu x, y) x, t) dt, x x y zu x zu x, y) s, y) ds, y x y for u U,, 1,.... x, y) a.e., x, y) a.e., As an immediate consequence of Lemma 3, we obtain the following result: Corollary 4. The set {zu AC, R n ) ) : u U,, 1,... } is bounded in AC, R n ).

12 4 Lemma 4. U U in L, R m ) with respect to the Hausdorff metric. roof. Fix ε >. Let N be such that M NM, ε) and M NM, ε) for. This implies that, for any fixed function u U ), distu x, y), M ) ε, x, y) a.e. Using the theorem on the measurable selection Kisielewicz, 1991, Thm. 3.13), we conclude that there exists a function u U such that u x, y) u x, y) ε, x, y) a.e., that is, u u L ) ε. Therefore, U NU, ε) for. In the same way, we chec that U NU, ε) for. Lemma 5. For each ε >, there exist N and δ, ε) such that U NU, δ) and U NU, δ) for, and if u U ), u U are such that u u L ) < δ, then z u z u o AC,Rn ) < ε. roof. Let us observe that if u U, u U z u AC, R n ) is a solution of the system z x y f x, y, z, z x, z y, u), satisfying the zero-boundary conditions, and zu AC, R n ) is a solution of the system z x y f x, y, z, z x, z y, u), which also satisfies the zero-boundary conditions, then g u z u / x y is a fixed point of the operator F u and gu z u / x y is a fixed point of the operator Fu. Moreover, g u g l u F u g u ) F u g u ) l F u g u ) F u g u ) l F u g u ) F u g u ) l α g u g u l F u g u ) F u g u ) l. Hence D. Idcza et al. g u g l u 1 F u 1 α g u ) F u g u ) l. Since g u g u e l g u g l u, F u g u ) F u g u ) l F u g u ) F u g u ), we obtain z u z AC u ) g u g u el F u 1 α g u ) F u g u ) el F u 1 α g u ) F u g u ) el F u 1 α g u ) F u g u ) el 1 α 1 1 f x, y, el 1 α f x, y, 1 1 y gu, β x, y, y x ) gu, gu, gu, u x, y) x gu, ) ) 1 gu, u x, y) dx dy y gu, u x, y) u x, y) ) 1 dx dy. x gu, g u ) N ow, let us fix ε > and choose δ, ε) to be such that e l ) 1 α el 1 α γ C L,R ) δ < ε C is the ball in R n R n R n centred at with radius 3r, where r is described in Lemma 3). Let 1 N be such that cf. Lemma 4) U NU, δ) and U NU, δ) for 1, and let satisfy cf. Assumption N4)) for. ϕ C L 1,R ) < δ Set max{ 1, } and fix. If u U, u U are such that u u L,R m ) < δ, then

13 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem 41 we obtain z u z u AC,R n ) el 1 α δ el 1 α γ C L,R ) δ < ε and the proof is complete. In the sequel, we shall assume that each roblem NH ),, 1,... has a solution. As in the previous part, the set of all solutions to roblem NH ) will be denoted by A. The minimal value of roblem NH ) will be denoted by m. By an upper limit of sets A AC, R n ) L, R m ) we mean the set lim sup A of all cluster points in the space AC, R n ) L, R m )) of sequences z u, u )) N, where z u u, ) A. Now, we shall prove the main result of this part: Theorem 4. If Assumptions N1) N7) are satisfied and the sets A,, 1,..., are non-empty, then a) m m, b) lim sup A A in the space AC, R n ) L, R m )). roof. Fix η >. From the Scorza-Dragoni Theorem Eeland and Temam, 1976), applied to the function F C M C is described in the proof of Lemma 5), it follows that there exists a compact set η such that µ \ η ) η the Lebesgue measure) and F η C M is uniformly continuous. In particular, this means that there exists σ > such that F x, y, z, u) F x, y, w, υ) < η, provided that z w, u υ < σ, x, y) η. Now, for any positive integer, we fix a pair z u, u ) A. Let, δ be the constants from Lemma 5 applied to ε σ. Then, for each, there exists u ) U such that u x, y) u ) x, y) < σ, x, y) a.e., and z u x, y) z u )x, y) z u z u ) Consequently, F x, y, z u x, y), u x, y) ) for. AC ) < σ, F x, y, z u )x, y), u ) x, y)) < η, x, y) η x, y). a.e. Thus J z u, u ) J z u u ), ) ) η F x, y, z u x, y), u x, y) ) F x, y, z u )x, y), u ) x, y)) dx dy \ η ) F x, y, z u x, y), u x, y) F x, y, z u )x, y), u ) x, y)) dx dy µ η )η \ η ν B x, y) dx dy for B is the ball in R n centred at with radius r described in Lemma 3). Writing η η \ η ν B x, y) dx dy, we have m J z u, u ) J z u ), u ) ) J z u, u ) η for. From Assumption N7) it follows that there exists such that J z u, u ) J z u, u ) η for. So, for max{, }, we have m J z u, u ) η m η. In the same way, we show that m m η for sufficiently large parameters. Indeed, for each, there exists u ) U such that u x, y) u ) x, y) < σ, x, y) a.e., and zu x, y) z u y) )x, z u z u < σ, ) AC ) Consequently, J z u, u ) J z u ), u ) ) η for and m J z u, u ) J z u ), u ) ) J z u ), u ) ) η J z u, u ) η x, y).

14 4 for max{, }, which completes the proof of a) we have used here the fact that if l L 1 ), then the set function S l dµ is absolutely continuous, i.e. S l dµ as µs) ). S Now, assume that z, u) is a cluster point in AC, R n ) L, R m ) of some sequence z u, u )) N such that z u u, ) A. Without loss of generality, we may assume that z u in AC z, R n ) and u u in L, R m ) as. We shall show that z, u) A, i.e. u U, z zu and m J z, u). Fix ε >. Assumption N1) implies that there exists such that for, i.e. u x, y) NM, ε), x, y) a.e. u U NM,ε) { u L, R m ); ux, y) NM, ε), Since M is compact, so is NM, ε). Thus In particular, ux, y) NM, ε), ux, y) NM, 1/n), x, y) a.e. }. x, y) a.e. x, y) a.e. for n 1,,.... Fix a point x, y) which satisfies the above relation for n 1,,... of course, the set of such points has a full measure). There exists a sequence v n ) n N of points belonging to M, such that ux, y) v n 1 n for n 1,,.... In other words, ux, y) lim n v n, i.e. ux, y) M by the closedness of M ). Now, we shall prove that z z u. Indeed, z u x, y) f s, t, z u s, t), z u s, t), x z ) u y s, t), u s, t) ds dt f s, t, z u s, t), z u s, t), x z ) u y s, t), u s, t) f s, t, z u s, t), z u s, t), x z ) ) u y s, t), u s, t) ds dt f s, t, z u s, t), z u s, t), x z ) u y s, t), u s, t) ds dt for x, y). Consequently, zx, y) lim z u x, y) D. Idcza et al. f s, t, zs, t), z s, t), x z ) y s, t), us, t) ds dt f s, t, zs, t), z z ) s, t), x y s, t), us, t) ds dt for x, y). This means that z AC, R n ) and z x, y) x y f x, y, zx, y), z z ) x, y), x y x, y), ux, y) for x, y) a.e. To complete the proof, it suffices to show that m J z, u). We have m lim m lim J z u, u ) lim 1 1 lim 1 1 F x, y, z u x, y), u x, y) ) dx dy F x, y, z u x, y), u x, y) ) F x, y, z u x, y), u x, y) )) dx dy lim 1 1 F x, y, z u x, y), u x, y) ) dx dy.

15 Stability analysis of solutions to an optimal control problem associated with a Goursat-Darboux problem 43 The first term equals zero in view of Assumption N7). The other is equal to 1 1 F x, y, zx, y), ux, y)) dx dy on the basis of a generalization of the Krasnoselsii theorem Idcza and Rogowsi, 3). Hence m 1 1 F x, y, zx, y), ux, y))dx dy J z, u), which completes the proof Nonhomogeneous roblem Now, consider the family of problems w x y g x, y, w, w x, w ) y, u, x, y) a.e., wx, ) ϕ x), w, y) ψ y), x, y [, 1], ϕ ) ψ ) c, I w, u) 1 1 G x, y, w, u) dx dy min, u U { u L, R m ); ux, y) M, x, y) a.e. },, 1,.... By using the substitution N ) zx, y) wx, y) ϕ x) ψ y) c, x, y), it is easy to see that roblems N ) and NH ) with f x, y, z, z x, z y, u) g x, y, z ϕ x) ψ y) c, z x. ϕ x), z y. ψ y), u ), F x, y, z, u) G x, y, z ϕ x) ψ y) c, u ) are equivalent, i.e. w AC, R n ) is a solution to roblem N ) if and only if z AC, R n ), given by the above substitution, is a solution to roblem NH ),, 1,.... Moreover, the minimal values of functionals I w, u) and J z, u) are the same. Next, by C 1 [, 1], R m ) we mean the space of continuously differentiable functions ω : [, 1] R n, with the norm given by ω C1 [,1] max { ωt) ; t [, 1] } max {. ωt), t [, 1] }. Theorem 5. If the functions g, G,, 1,..., and the sets M,, 1,..., satisfy Assumptions N) N7), the sets B,, 1,... of solutions to roblems N ) are non-empty and the functions ϕ, ψ tend to ϕ, ψ, respectively, in the space C 1 [, 1], R m ), then a) m m, b) lim sup B B in the space AC, R n ) L, R m )). roof. It is easy to show that the functions f, F given above satisfy Assumptions N) N7). art a) of the assertion is obvious. To prove art b), assume that w u u, ) B, 1,,... and w u, u ) w, u) in AC, R n ) L, R m ). Then u U, as shown in the proof of Theorem 4. Moreover, z u, u ) A, 1,,..., where and z u x, y) w u x, y) ϕ x) ψ y) c, x, y), z y u, u ) z, u) in AC, R n ) L, R m ), where zx, y) wx, y) ϕ x) ψ y) c, x, y). Consequently, Theorem 4 implies z, u) A. means that w, u) B. References This Aubin J.. and Franowsa H. 199): Set-Valued Analysis. Boston: Birhäuser. Ban B., Guddat J., Klatte D., Kummer B. and Tammer K. 1983): Non-Linear arametric Optimization. Basel: Birhäuser. Bergmann J., Burgemeier. and Weiher. 1989): Explicit controllability criteria for Goursat problem. Optim., Vol., No. 3, pp Eeland I. and Temam R. 1976): Convex Analysis and Variational roblems. Amsterdam: North-Holland ublishing Company. Fiacco A.V. 1981): Sensitivity analysis for nonlinear programming using penalty methods. Math. rogramm., Vol. 1, No. 3, pp

16 44 D. Idcza et al. Idcza D. 199): Optimization of systems described by partial differential equations. h. D. Thesis, University of Łódź, oland, in olish). Idcza D. 1996): The generalization of the Du Bois-Reymond lemma for functions of two variables to the case of partial derivatives of any order. Topology in Nonlinear Analysis, Banach Center ublications, Vol. 35, pp Idcza D. 1998): Optimal control of a coercive Dirichlet roblem. SIAM J. Contr. Optim., Vol. 36, No. 4, pp Idcza D. and Rogowsi A., 3): On a generalization of Krasnoselsii s theorem. J. Austral. Math. Soc., Ser. A, to appear). Idcza D. and Walcza S. 1994): On Helly s theorem for functions of several variables and its applications to variational problems, Optim., Vol. 3, No. 4, pp Idcza D., Kibalczyc K. and Walcza S. 1994): On an optimization problem with cost of rapid variation of control. J. Austral. Math. Soc., Ser. B, Vol. 86, No.1, pp Ioffe A. 1994): On sensitivity analysis of nonlinear programs in Banach spaces: the approach via composite unconstrained optimization. SIAM J. Optim., Vol. 4, No. 1, pp Kisielewicz M., 1991): Differential Inclusions and Optimal Control. Dordrecht-Warsaw: Kluwer Academic ublishers olish Scientific ublishers. Levitin E.S. 1975): On the local perturbation theory of a problem of mathematical programming in a Banach space. Soviet Math. Dol., Vol. 16, No. 5, pp Łojasiewicz S. 1988): An Introduction to the Theory of Real Functions. Chichester: Wiley. Malanowsi K. 1993): Stability and sensitivity analysis of solutions to infinite-dimensional optimization problems. roc. 16-th IFI Conf. System Modelling and Optimization, Compiegne, France, pp ulvirenti G. and Santagati G. 1975): Sulla controllabilita completa in Luogo determinato ed in luogo variabile por un processo di controllo con parametri dostributi. Boll. UMI, Vol. 11, No. 3, pp , in Italian). Robinson S.M. 1974): erturbed Kuhn-Tucer points and rates of convergence for a class of nonlinear systems. Math. rogramm., 7, pp Robinson S.M. 1976): Stability theory for system of inequalities, art II: Differentiable nonlinear systems. SIAM J. Numer. Anal. Vol. 13, No. 4, pp Robinson S.M. 198): Strongly regular generalized equations. Math. Oper. Res., Vol. 5, No. 1, pp Surryanarayama M.B. 1973): Necessary conditions for optimization problems with hyperbolic partial differential equations. SIAM J. Contr., Vol. 11, No. 4, pp Tihonov A.N. and Samarsi A.A. 1958): Equations of Mathematical hysics. Moscow: GITTL, in Russian). Walcza S. 1987): Absolutely continuous functions of several variables and their applications to differential equations. Bull. ol. Acad. Sci., Ser. Math., Vol. 35, No. 11 1, pp Walcza S. 1): Well-posed and ill-posed optimal control problems. JOTA, Vol. 19, No. 1, pp