Weak Singular Optimal Controls Subject to State-Control Equality Constraints

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 212, no. 36, Weak Singular Optimal Controls Subject to State-Control Equality Constraints Javier F Rosenblueth IIMAS UNAM, Universidad Nacional Autónoma de México Apartado Postal 2-726, México DF 1 jfrl@unam.mx Gerardo Sánchez Licea Facultad de Ciencias, Universidad Nacional Autónoma de México Departamento de Matemáticas, México DF 451 gslicea@apolo.acatlan.unam.mx Abstract The aim of this paper is to derive and illustrate a new set of second order sufficient conditions for weak minima of optimal control problems involving mixed (state-control) equality constraints. The conditions include the Legendre-Clebsch necessary condition but not its strengthened form, as well as a new version of the strengthened Weierstrass condition. Being nonsingularity crucial in the literature for sufficient results, the possible singularity of the weak optimal control in the main theorem of the paper renders sufficiency theory a much wider scope. Mathematics Subject Classification: 49K15 Keywords: Optimal control, sufficient conditions for optimality, weak minima, singularity, mixed equality constraints 1 Introduction In this paper we shall be concerned with the fixed-endpoint optimal control problem, which we label (P), of minimizing the functional I(x, u) := L(t, x(t),u(t))dt over all couples x:[,t 1 ] R n absolutely continuous and u:[,t 1 ] R m essentially bounded, satisfying the constraints

2 1798 J.F. Rosenblueth and G. Sánchez Licea a. ẋ(t) =f(t, x(t),u(t)) a.e. in [,t 1 ], b. x( )=ξ, x(t 1 )=ξ 1, c. φ(t, x(t),u(t)) = (t [,t 1 ]), where the interval T := [,t 1 ] is fixed, and we are given two points ξ and ξ 1 R n, and functions L, f and φ mapping T R n R m to R, R n and R q respectively. Second order sufficient conditions for this kind of problems are well-known in the literature (see, for example, [3, 4] and references therein). The main approaches correspond to finding a bounded solution to a matrix-valued Riccati equation, or the insertion of the original problem as an optimization problem in a Banach space, and both require that the strengthened Legendre-Clebsch condition holds. In recent papers [6, 8] we derived a new set of sufficient conditions for problems involving equality constraints only in the control functions (not mixed with the state) without assuming the strengthened Legendre-Clebsch condition. Thus, the set of sufficient conditions obtained in those papers does not require the nonsingularity of the process under consideration. In [6], the main result corresponds to sufficient conditions for a weak minimum and, in [7], we provide some examples and applications. In [8], the conditions obtained are sufficient for a strong minimum. In this paper we generalize the results of [6] for weak minima involving mixed equality constraints. The technique used is similar to that of weak minima for problems involving control equality constraints, but some crucial differences arise in the insertion of the state variables. We also illustrate the main result of the paper by providing two examples that lie beyond the scope of [3, 4], due to the singularity of the weak optimal control, and of [6], due to the presence of mixed constraints. 2 Notation and the main results For simplicity of notation, let us denote by X the space of absolutely continuous functions mapping T to R n, and set U r := L (T ; R r )(r N). Elements of X U m are called processes and a process is admissible if it satisfies the constraints. An admissible process (x, u) is called a weak minimum of (P) if, for some ɛ>, I(x, u) I(y, v) for all admissible process (y, v) not equal to (x, u) and satisfying (y, v) (x, u) <ɛ. If the inequality can be replaced by a strict inequality, the minimum is said to be a strict weak minimum. We shall assume throughout the paper that the functions L, f and φ are C 2 with respect to x and u on T R n R m.

3 Weak singular optimal controls 1799 Let us consider the Hamiltonian function H given by H(t, x, u, p, μ) := p, f(t, x, u) L(t, x, u) μ, φ(t, x, u) where p R n denotes the adjoint variable and μ R q is the multiplier associated with the mixed state-control constraint φ(t, x(t),u(t)) =. Under normality assumptions (see [2]) first order necessary conditions state that, if (x, u) is a weak minimum of (P), then there exist p X and μ U q such that ṗ(t) = Hx (t, x(t),u(t),p(t),μ(t)) a.e. in T, H u (t, x(t),u(t),p(t),μ(t)) = (t T ) where denotes transpose. Usually the term extremal corresponds to admissible processes (x, u) for which the existence of p and μ satisfying those relations can be assured. We shall find convenient to express the sufficient conditions in terms of the following function associated with the Hamiltonian. Given p X and μ U q define, for all (t, x, u) T R n R m, F (t, x, u) :=L(t, x, u) p(t),f(t, x, u) + μ(t),φ(t, x, u) ṗ(t),x. Note that F (t, x, u) = H(t, x, u, p(t),μ(t)) ṗ(t),x. Following the definition of nonsingularity given by Hestenes in [2], a process (x, u) will be called nonsingular if the determinant F uu (t, x(t),u(t)) is different from zero (t T ). With respect to F (which depends on p and μ), let J(x, u) := p(t 1 ),ξ 1 p( ),ξ + F (t, x(t),u(t))dt. Consider the first variation of J along (x, u) X U m over (y, v) X L 1 (T ; R m ) given by J ((x, u); (y, v)) := {F x (t, x(t),u(t))y(t)+f u (t, x(t),u(t))v(t)}dt and the second variation of J along (x, u) X U m over (y, v) X L 2 (T ; R m ) given by J ((x, u); (y, v)) := 2Ω(t, y(t),v(t))dt, where, for all (t, y, v) T R n R m, 2Ω(t, y, v) := y, F xx (t, x(t),u(t))y +2 y, F xu (t, x(t),u(t))v + v, F uu (t, x(t),u(t))v.

4 18 J.F. Rosenblueth and G. Sánchez Licea Finally, for all u L 1 (T ; R m ) let D(u) := ϕ(u(t))dt where ϕ(c) :=(1+ c 2 ) 1/2 1. The following theorem corresponds to the main result of the paper, a sufficiency result for a strict weak minimum of problem (P). It should be noted that the conditions include the Legendre-Clebsch necessary condition, the positivity of the second variation on a certain set of admissible variations, and a condition related to the Weierstrass excess function of the auxiliary function F. Theorem 2.1 Let (x,u ) be an admissible process. p X, μ U q, and h, ɛ > such that Suppose there exist ṗ(t) = H x(t, x (t),u (t),p(t),μ(t)) a.e. in T, H u (t, x (t),u (t),p(t),μ(t)) = (t T ), and the following conditions are satisfied: i. F uu (t, x (t),u (t)) (a.e. in T ). ii. J ((x,u ); (y, v)) > for all nonnull (y, v) X L 2 (T ; R m ) satisfying a. ẏ(t) = f x (t, x (t),u (t))y(t) +f u (t, x (t),u (t))v(t) a.e. in T, and y( )=y(t 1 )=; b. φ x (t, x (t),u (t))y(t)+φ u (t, x (t),u (t))v(t) =a.e. in T. iii. For all admissible processes (x, u) satisfying (x, u) (x,u ) <ɛ, E(t, x(t),u (t),u(t))dt hd(u u ) where E denotes the Weierstrass excess function with respect to F, E(t, x, u, v) :=F (t, x, v) F (t, x, u) F u (t, x, u)(v u). Then there exist ρ, δ > such that, for all admissible processes (x, u) satisfying (x, u) (x,u ) <ρ, I(x, u) I(x,u )+δd(u u ). In particular, (x,u ) is a strict weak minimum of (P). Let us end this section by deriving a simple consequence of Theorem 2.1. It corresponds to a sufficiency result which, in certain cases, may be much easier to verify than the previous one, though both the continuity of the control and the nonsingularity of the process under consideration are assumed.

5 Weak singular optimal controls 181 Corollary 2.2 Let (x,u ) be an admissible process with u Suppose there exist p X and μ U q such that continuous. ṗ(t) = H x(t, x (t),u (t),p(t),μ(t)) a.e. in T, H u (t, x (t),u (t),p(t),μ(t)) = (t T ), and the following conditions are satisfied: i. F uu (t, x (t),u (t)) > (t T ). ii. J ((x,u ); (y, v)) > for all nonnull (y, v) X L 2 (T ; R m ) satisfying a. ẏ(t) = f x (t, x (t),u (t))y(t) +f u (t, x (t),u (t))v(t) a.e. in T, and y( )=y(t 1 )=; b. φ x (t, x (t),u (t))y(t)+φ u (t, x (t),u (t))v(t) =a.e. in T. Then there exist ρ, δ > such that, for all admissible processes (x, u) satisfying (x, u) (x,u ) <ρ, I(x, u) I(x,u )+δd(u u ). In particular, (x,u ) is a strict weak minimum of (P). Proof: Define a restricted tube of radius ɛ> centred on (x,u )as T 1 ((x,u ); ɛ) :={(t, y, v) T R n R m : x (t) y <ɛ, u (t) v <ɛ}. By 2.2(i) and since u is continuous, there exist h, ɛ > such that c, F uu (t, x, u)c h c 2 (c R m, (t, x, u) T 1 ((x,u ); ɛ)). Hence, for (t, x, u, v) with (t, x, u) and (t, x, v) int 1 ((x,u ); ɛ), E(t, x, u, v) = (1 λ) v u, F uu (t, x, u + λ[v u])(v u) dλ 1 2 h v u 2 hϕ(v u). The conclusion follows by Theorem Proof of Theorem 2.1 In this section we shall prove Theorem 2.1. We first state an auxiliary result which is an immediate consequence of Lemma 3.1 of [5] and on which the proof of the theorem is strongly based.

6 182 J.F. Rosenblueth and G. Sánchez Licea Lemma 3.1 Let {u q } be a sequence in L 1 (T ; R m ), u L 1 (T ; R m ), and suppose that lim D(u q u )= and d q := [2D(u q u )] 1/2 > (q N). For all q N and t T define w q (t) := [ 1+ 1 ] 1/2, 2 ϕ(u q(t) u (t)) vq (t) := u q(t) u (t). d q Then the following hold: a. For some v L 2 (T ; R m ) and some subsequence of {u q }, again denoted by {u q }, {v q } converges weakly to v in L 1 (T ; R m ). b. Let A q L (T ; R n n ) and B q L (T ; R n m ) be matrix functions for which there exist constants m,m 1 > such that A q m, B q m 1 (q N), and denote by y q the solution of the initial value problem ẏ(t) =A q (t)y(t)+b q (t)v q (t) (a.e. in T ), y( )=. Then there exist σ L 2 (T ; R n ) and a subsequence of {y q }, again denoted by {y q }, such that {ẏ q } converges weakly in L 1 (T ; R n ) to σ. Moreover, if y (t) := t σ (s)ds (t T ), then y q (t) y (t) uniformly on T. c. Suppose that w q (t) 1 uniformly on T, let R q ( ) be an m m real matrix-valued measurable function on T, R ( ) L (T ; R m m ), and assume also that R q (t) R (t) uniformly on T, and R (t) a.e. in T. Then, for some subsequence of {u q }, again denoted by {u q }, lim inf R q (t)v q (t),v q (t) dt R (t)v (t),v (t) dt. Proof of Theorem 2.1: Assume that, for all ρ, δ >, there exists an admissible process (x, u) with (x, u) (x,u ) <ρsuch that J(x, u) <J(x,u )+δd(u u ). (1) We shall show that this contradicts 2.1(ii) and the statement will follow, since J(x, u) = I(x, u) for all admissible processes (x, u). Let z := (x,u ). Note that, for all admissible processes z =(x, u), J(z) =J(z )+J (z ; z z )+K(z)+Ẽ(z) (2) where Ẽ(x, u) := E(t, x(t),u (t),u(t))dt,

7 Weak singular optimal controls 183 K(x, u) := {M(t, x(t)) + u(t) u (t),n(t, x(t)) }dt, and the functions M and N are given by M(t, y) :=F (t, y, u (t)) F (t, x (t),u (t)) F x (t, x (t),u (t))(y x (t)), N(t, y) :=Fu (t, y, u (t)) Fu(t, x (t),u (t)). By Taylor s theorem we have M(t, y) = 1 2 y x (t),p(t, y)(y x (t)), N(t, y) =Q(t, y)(y x (t)), where P (t, y) :=2 (1 λ)f xx (t, x (t)+λ(y x (t)),u (t))dλ, Q(t, y) := F ux (t, x (t)+λ(y x (t)),u (t))dλ. Now, by (1), for all q N there exists an admissible process z q := (x q,u q ) such that z q z < 1 q, J(z q) J(z ) < 1 q D(u q u ). (3) Since z q is admissible, observe that the last inequality implies that u q (t) u (t) on a set of positive measure and so D(u q u ) > (q N). Since z q z asq, it follows that D(u q u ), q. Define d q, w q, v q as in Lemma 3.1 and, for all t T and q N, set y q (t) := x q(t) x (t) d q. By Lemma 3.1(a) there exist v L 2 (T ; R m ) and some subsequence of {z q } (we do not relabel) such that {v q } converges weakly in L 1 (T ; R m )tov. By Taylor s theorem, for all q N we have ẏ q (t) =A q (t)y q (t)+b q (t)v q (t) (a.e. in T ) where A q (t) = B q (t) = f x (t, x (t)+λ[x q (t) x (t)],u (t))dλ, f u (t, x q (t),u q (t)+λ[u (t) u q (t)])dλ. By continuity of f x and f u there exist m,m 1 > such that A q m and B q m 1 (q N). By Lemma 3.1(b) there exist σ L 2 (T ; R n ) and some subsequence of {z q } (we do not relabel) such that, if y (t) := t σ (s)ds (t T ), then y q (t) y (t) uniformly on T.

8 184 J.F. Rosenblueth and G. Sánchez Licea The theorem will be proved if we show that J (z ;(y,v )), (y,v ) (, ), y ( )=y (t 1 ) =, and the following two relations hold: ẏ (t) =f x (t, x (t),u (t))y (t)+f u (t, x (t),u (t))v (t) a.e. in T, φ x (t, x (t),u (t))y (t)+φ u (t, x (t),u (t))v (t) = a.e. in T. To prove it observe that, since z q is admissible, the fact that y ( ) = y (t 1 ) = follows by Lemma 3.1(b). Now, by definition of the functional K, for all q N, K(z q ) d 2 q = In view of Lemma 3.1(b), { M(t, xq (t)) + d 2 q v q (t), N(t, x q(t)) d q } dt. M(t, x q (t)) lim = 1 d 2 q 2 y (t),f xx (t, x (t),u (t))y (t), N(t, x q (t)) lim = F ux (t, x (t),u (t))y (t) d q both uniformly on T and, since {v q } converges weakly to v in L 1 (T ; R m ), 1 2 J K(z q ) (z ;(y,v )) = lim + 1 v d 2 (t),f uu (t, x (t),u (t))v (t) dt. (4) q 2 Now, by Taylor s theorem, for all t T and q N, where Clearly, 1 E(t, x q (t),u (t),u q (t)) = 1 2 v q(t),r q (t)v q (t) d 2 q R q (t) :=2 (1 λ)f uu (t, x q (t),u (t)+λ[u q (t) u (t)])dλ. lim R q(t) =R (t) :=F uu (t, x (t),u (t)) uniformly on T. Since z q z asq, it follows that w q (t) 1 uniformly on T and by 2.1(i), R (t) a.e. in T. By Lemma 3.1(c), there is a subsequence of {z q } (we do not relabel) such that lim inf Ẽ(z q ) d 2 q 1 2 v (t),f uu (t, x (t),u (t))v (t) dt. (5)

9 Weak singular optimal controls 185 Since ṗ(t) = Hx (t, x (t),u (t),p(t),μ(t)) a.e. in T, H u (t, x (t),u (t),p(t),μ(t)) = (t T ), it follows that J (z ;(y, v)) = for all (y, v) X L 1 (T ; R m ). With this in mind, (2), (3), (4) and (5), 1 2 J K(z q ) Ẽ(z q ) (z ;(y,v )) lim + lim inf d 2 q d 2 q = lim inf J(z q ) J(z ) d 2 q. In addition, if (y,v )=(, ), then lim K(z q )/d 2 q = and, by 2.1(iii), h 2 lim inf Ẽ(z q ) d 2 q contradicting the positivity of h. Now, observe that y q (t) y (t), A q (t) f x (t, x (t),u (t)), B q (t) f u (t, x (t),u (t)) all uniformly on T, and {v q } converges weakly to v in L 1 (T ; R m ). Therefore {ẏ q } converges weakly in L 1 (T ; R n )to f x (t, x (t),u (t))y (t)+f u (t, x (t),u (t))v (t). By Lemma 3.1(b), {ẏ q } converges weakly in L 1 (T ; R n )toσ =ẏ. Hence, ẏ (t) =f x (t, x (t),u (t))y (t)+f u (t, x (t),u (t))v (t) a.e. in T. Finally to show that φ x (t, x (t),u (t))y (t)+φ u (t, x (t),u (t))v (t) = a.e. in T, for all q N, t T and λ [, 1], set G q (t; λ) :=φ(t, x (t)+λ[x q (t) x (t)],u (t)+λ[u q (t) u (t)]). Clearly, for all q N and t T, By Taylor s theorem, if we set and G q (t;)=g q (t;1)=. φ x (t; λ; q) :=φ x (t, x (t)+λ[x q (t) x (t)],u (t)+λ[u q (t) u (t)]) φ u (t; λ; q) :=φ u (t, x (t)+λ[x q (t) x (t)],u (t)+λ[u q (t) u (t)]),

10 186 J.F. Rosenblueth and G. Sánchez Licea it follows that for all q N and t T, ( ) ( ) φ x (t; λ; q)dλ y q (t)+ φ u (t; λ; q)dλ v q (t) =. Since z q z, y q (t) y (t) uniformly on T and {v q } converges weakly to v in L 1 (T ; R m ), t {φ x (s, x (s),u (s))y (s)+φ u (s, x (s),u (s))v (s)}ds = for all t T. Consequently, φ x (t, x (t),u (t))y (t)+φ u (t, x (t),u (t))v (t) = a.e. in T and this completes the proof. 4 Examples In this section we provide two examples of a fixed-endpoint optimal control problem with equality state-control constraints for which an application of Theorem 2.1 shows that the singular extremal under consideration is in fact a strict weak minimum. Example 4.1 Consider the problem of minimizing I(x, u) = ( t/2)x(t)[u(t) 1]dt subject to a. ẋ(t) =( t/2)x(t)u(t) tx(t) a.e. in [, 1]. b. x() = 1, x(1) = e 2/3. c. x(t)u(t)e u(t) ax 2 (t)u(t) =(t [, 1]). For this case, n = m = q =1,T =[, 1], ξ =1,ξ 1 = e 2/3,<a<1, L(t, x, u) =( t/2)x[u 1], f(t, x, u) =( t/2)xu tx, φ(t, x, u) =xue u ax 2 u. Let z := (x,u ) (e ( 2/3)t3/2, ) which is admissible. We have H(t, x, u, p, μ) =(p t/2)xu p tx +( t/2)x[1 u] μ[xue u ax 2 u], H x (t, x, u, p, μ) =(p t/2)u p t +( t/2)[1 u] μ[ue u 2axu], H u (t, x, u, p, μ) =(p t/2)x ( t/2)x μ[xe u xue u ax 2 ].

11 Weak singular optimal controls 187 Let p 1/2 and t μ(t) := 4[1 ae ( 2/3)t3/2 ] (t T ). As one readily verifies (x,u,p,μ) satisfies the first order conditions of Theorem 2.1. Moreover, the function F is given by and, hence, F (t, x, u) = txu 4 t[xue u ax 2 u] 4[1 ae ( 2/3)t3/2 ] F uu (t, x (t),u (t)) = te ( 2/3)t 3/2 2[1 ae ( 2/3)t3/2 ] (t T ) implying that z is singular and the first condition of Theorem 2.1 holds. Also, observe that f x (t, x (t),u (t)) = te ( 2/3)t 3/2 t and f u (t, x (t),u (t)) = (t T ) 2 and hence (y, v) satisfies (a) and (b) of the second condition if and only if ẏ(t) = te ( 2/3)t 3/2 ty(t)+ v(t) a.e. in T, 2 y() = y(1) =, and e ( 2/3)t3/2 [1 e ( 2/3)t3/2 ]v(t) = a.e. in T. Consequently, there are no nonnull admissible variations and the second condition is fulfilled. Now observe that, by Taylor s theorem, ( ) E(t, x,,u)= (1 λ)f uu (t, x, λu)dλ u 2, and txg(u) F uu (t, x, u) = 4[1 ae ( 2/3)t3/2 ], where g(u) :=2e u ue u. Hence, since g() = 2 > and x (t) =e ( 2/3)t3/2 > (t T ), there are two positive numbers δ, ɛ such that g(w) δ for all w ( ɛ, ɛ), and T (x ; ɛ) :={(t, x) T R : x x (t) <ɛ}

12 188 J.F. Rosenblueth and G. Sánchez Licea is contained in Q δ := {(t, x) R 2 t, x δ}. Thusif(t, x, u) belongs to then T 1 ((x,u ); ɛ) ={(t, x, u) T (x ; ɛ) R : u u (t) <ɛ}, ( E(t, x,,u)= (1 λ) ) txg(λu) 4[1 ae ( 2/3)t3/2 ] dλ u 2 δ2 t 8[1 a] u2. Let us now suppose that the third condition does not hold. Then, for all q N, there exists an admissible process z q := (x q,u q ) such that z q z < 1/q, z q z <ɛ, and E(t, x q (t),u (t),u q (t))dt < 1 q D(u q). Since for all q N and almost all t T, E(t, x q (t),u (t),u q (t)) δ2 t 8[1 a] u2 q (t), for all q N we have δ 2 v t q 2(t) 8[1 a] wq 2(t)dt δ 2 tv 2 8[1 a] q (t)dt < 1 2q where w q (t) :=[ ϕ(u q(t))] 1/2, v q (t) := u q(t) d q, and d q := [2D(u q )] 1/2. From the above we conclude that lim Now note that, for all q N, v 2 t q (t) dt =. wq(t) 2 By (6) and integrating by parts, for all q N, v t q 2(t) ( t wq 2(t)dt + vq 2(s) ) dt wq 2(s)ds 2 t vq(t) 2 =1. (6) wq 2 (t)dt =1. (7) Since h q (t) := t v2 q (s)/w2 q (s)ds 1(t T, q N), by (7), for all q N, v 2 q(t) 1 t wq 2(t)dt + h 2 q(t) dt 2 t 1. (8)

13 Weak singular optimal controls 189 By (8), there exist some subsequence (we do not relabel) and some h in L 2 (T ; R) such that the sequence α q (t) :=h q (t)/ 4 t converges weakly to h in L 2 (T ; R), that is, h q (t) 1 lim 4 g(t)dt = h (t)g(t)dt for all g L 2 (T ; R). (9) Set g(t) :=1/ 4 t if <t 1, and g() :=. Since 1 tv 2 q (t)/wq 2 (t)dt as q, by using g given above, it follows by (7) and (9) that h (t) 4 t dt =2. (1) We claim that h (t) = 1/ 4 t a.e. in T. To prove it, observe first that h (t) a.e. in T for, otherwise, there would exist S subset of T such that m(s) > and h (t) < (t S). Defining g(t) :=1ift S and g(t) :=if t T \ S and replacing g in (9) by g, we obtain h q (t) lim 4 dt = h (t)dt < S t S which is a contradiction. Now, observe that h (t) dt = h 2 t (t)dt 2 Once again by (9) and (1), h (t) 1 4 dt + h 2 (t)dt = lim h q (t) h (t) 1 h (t) 4 dt t 4 dt =2, t dt 1 = h 2 (t)dt 2. and this proves our claim. By (9), it is readily seen that lim h q (t)g(t)dt = g(t)dt for all g L 2 (T ; R), that is, h q converges weakly to 1 in L 2 (T ; R). Since lim sup h q 2 1 2, by Proposition 3.32 of [1, p. 78], we have h q 1 2 as. Then by (6), = lim t v 2 q(s) w 2 q (s)ds 1 2 dt = lim t and hence for some subsequence (we do not relabel), lim t vq(s) 2 = pointwisely a.e. in T. wq 2 (s)ds v 2 q(s) w 2 q (s)ds 2 dt,

14 181 J.F. Rosenblueth and G. Sánchez Licea Consequently, lim vq(s) 2 = wq 2 (s)ds which contradicts (6). Therefore also the third condition holds and, by Theorem 2.1, (x,u ) is a strict weak minimum of problem (P). Example 4.2 Consider the problem of minimizing I(x, u) = {u 2 1(t) (1/2)u 2 2(t) (1/8)x(t)u 1 (t)}dt subject to a. ẋ(t) = sinh u 1 (t) (1/8)x 3 (t) a.e. in [, 1]. b. x() = x(1) =. c. x(t)u 2 1(t)+u 2 2(t) =4(t [, 1]). Note that, for this case, n = q =1,m =2,T =[, 1], ξ = ξ 1 =, L(t, x, u) =u 2 1 (1/2)u2 2 (1/8)xu 1, f(t, x, u) = sinh u 1 (1/8)x 3, φ(t, x, u) =xu u Let z := (x,u ) (,, 2). Clearly z is admissible, and we have H(t, x, u, p, μ) =p sinh u 1 (1/8)px 3 u 2 1 +(1/2)u2 2 +(1/8)xu 1 μxu 2 1 μu2 2 +4μ, H x (t, x, u, p, μ) = (3/8)px 2 +(1/8)u 1 μu 2 1, H u (t, x, u, p, μ) =(pcosh u 1 2u 1 +(1/8)x 2μxu 1,u 2 2μu 2 ). Therefore (x,u,p,μ) with (p, μ) (, 1/2) satisfies the first order conditions of Theorem 2.1. Now, the function F is given by Since, for all t T, F (t, x, u) =u 2 1 (1/8)xu 1 +(1/2)xu F uu (t, x (t),u (t)) = ( ) 2, it follows that F uu (t, x (t),u (t))h, h =2h 2 1, and so (x,u ) is singular and the first condition of Theorem 2.1 holds. Also, note that f x (t, x (t),u (t)) = and f u (t, x (t),u (t)) = (1, ) (t T ) so that, for any (y, v) satisfying (a) and (b) of the second condition, we must have ẏ(t) =v 1 (t) a.e. in T, y() = y(1) = and v 2 (t) = a.e. in T.

15 Weak singular optimal controls 1811 It follows that, for the second variation, for all (y, v) satisfying (a) and (b) of the second condition, J (z ;(y, v)) = {2v 2 1 (t) (1/4)y(t)v 1(t)}dt = 2v 2 1 (t)dt and the second condition of the theorem is satisfied. Suppose now that the third condition does not hold. Then, for all q N, there exists an admissible process z q := (x q,u q ) such that z q z < 1/q and E(t, x q (t),u (t),u q (t))dt < 1 q D(u q u ). Since E(t, x(t),u (t),u(t)) = E(t, x(t),, 2,u 1 (t),u 2 (t)) = u 2 1 (t)+(1/2)x(t)u2 1 (t), for all q N, u 2 1q(t)dt x q (t)u 2 1q(t)dt < 1 q D(u q u ). (11) Since x q (t)u 2 1q(t)+u 2 2q(t) =4,u 1q (t), u 2q (t) 2 both uniformly on T, ϕ(c) c 2 /2 and (2 + ϕ(c))ϕ(c) = c 2, we have the existence of some m > such that, for all q N, D(u q u ) 1 u q (t) u (t) 2 dt = 1 u 2 1q 2 2 (t)dt + 1 (u 2q (t) 2) 2 dt 2 = 1 u 2 1q 2 (t)dt + 1 x 2 q (t)u4 1q (t) 2 (2 + u 2q (t)) dt u 2 2 1q (t)dt u 2 1q(t) 2+ϕ(u 1q ) 2+ϕ(u 1q (t)) dt = 2+ϕ(u 1q ) ϕ(u 1q (t))dt m D(u 1q ). By (11) and since x q (t)u 2 1q (t) +u2 2q (t) =4(t T, q N), it follows that u 1q implying that D(u 1q ) > (q N). Consequently, D(u q u ) D(u 1q ) m (q N). Once again by (11), v1q(t) 2 w1q(t) dt + 1 w 2 2 1q(t)x 2 q (t) v2 1q(t) w1q(t) < m 2 2q (12)

16 1812 J.F. Rosenblueth and G. Sánchez Licea where, for all t T and q N, w 1q (t) :=[ ϕ(u 1q(t))] 1/2, v 1q (t) := u 1q(t) d 1q and d 1q := [2D(u 1q )] 1/2. But since for all q N, v1q 2 (t) dt =1, w1q(t) 2 w 1q (t) 1, x q (t) both uniformly on T, by letting q in (12), one obtains 1 which is a contradiction. Therefore also the third condition holds and, by Theorem 2.1, (x,u ) is a strict weak minimum of problem (P). References [1] Brezis H (21) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer [2] Hestenes MR (1966) Calculus of Variations and Optimal Control Theory, John Wiley, New York [3] Maurer H, Pickenhain S (1995) Second order sufficient conditions for control problems with mixed control-state constraints, Journal of Optimization Theory and Applications, 86: [4] Milyutin AA, Osmolovskiǐ NP (1998) Calculus of Variations and Optimal Control, Translations of Mathematical Monographs 18, Providence, Rhode Island: American Mathematical Society [5] Rosenblueth JF, Sánchez Licea G (21) A direct sufficiency proof for a weak minimum in optimal control, Applied Mathematical Sciences, 4: [6] Rosenblueth JF, Sánchez Licea G (211) Sufficiency for singular controls with equality constraints, Proceedings of the 13th IASTED International Conference on Intelligent Systems & Control, Cambridge, England, doi: /P , [7] Rosenblueth JF, Sánchez Licea G (in press) Singular weak optimal controls, Control & Intelligent Systems [8] Rosenblueth JF, Sánchez Licea G (in press) Sufficiency and singularity in optimal control, IMA Journal of Mathematical Control and Information Received: March, 212