On the Dirichlet problem for a Duffing type

We use direct variational method in order to investigate the dependence on parameter for the solution for a Duffing type equation with Dirichlet boundary value conditions.


Introduction
Recently the classical variational problem for a Duffing type equation received again some attention.In [1], [2], [7], some variational approaches were used in order to receive the existence of solutions for both periodic and Dirichlet type boundary value problems.Mainly direct method is applied under various conditions pertaining to at most quadratic growth imposed on the nonlinear term given in [2] and further relaxed in [7].Dirichlet problems for such equations could also be considered by some other methods, for example min-max theorem due to Manashevich, [8].In [6] the author gives some historical results concerning the Dirichlet problem for Duffing type equations and discusses the methods which are used in reaching the existence results which are different from the ones which we use and comprise the classical variational approach, the topological method.
In the boundary value problems for differential equations it is also important to know whether the solution, once its existence is proved, depends continuously on a functional parameter.This question has a great impact on future applications of any model since it is desirable to know whether the solution to the small deviation from the model would return, in a continuous way, to the solution of the original model.This is known in differential equation as stability or continuous dependence on parameter, see [4].We EJQTDE, 2011 No. 15, p. 1 will investigate the dependence on a functional parameter for a Duffing type equations basing on some results developed for different kind of problems in [4].However we provide some general principle which will allow for investigation of dependence on parameters for other problems also.In [4] and also in other papers by these authors, it is required that each problem should be investigated separately as far as the dependence on parameters is concerned.
Here we aim at providing some hint how to obtain a general rule, which will allow to investigate the dependence on parameters for various types of nonlinear problems.We will demonstrate our results on the Duffing type boundary value problem.
To be precise, in this paper we will consider the Dirichlet problem for a forced Duffing type equation with a functional parameter u.We investigate the problem with u : [0, 1] → R belonging to the set and where m > 0 is a fixed real number.Here f ∈ L 2 (0, 1) is the forcing term and r ∈ C 1 (0, 1) denotes the friction; r (τ ) ≥ 0 for τ ∈ [0, 1].Here we do not assume anything about the monotonicity of r, but instead we require that 1 4 r 2 (t) + 1 2 Of course, when r is nondecreasing we obviously have (2).Following [7] we denote R (t) = e R t 0 1 2 r(τ )dτ .Since r (τ ) ≥ 0 on [0, 1] we see that Upon putting y = R (t) x boundary problem (1) reads EJQTDE, 2011 No. 15, p. 2 Therefore instead of (1) in this paper we will investigate (4).In what follows (F 1 ) * denotes the Fenchel-Young transform (see for example [5]) of a function F 1 with respect to the second variable, namely As an application, we finally consider the existence to some optimal control problem.

The assumptions and examples
In order to apply a direct variational method to a Dirichlet problem (4) we will employ the following assumptions besides the assumptions given at the beginning of the paper.
F2 either t → (F 1 ) * (t, 0) is integrable on [0, 1] or else F 1 is convex in x for a.e.t ∈ [0, 1]; F3 there exist functions a, b ∈ L 2 (0, 1) such that With assumptions F1, F2, F3 we get for any fixed u ∈ L M the existence of an argument of a minimum for an Euler functional J u : EJQTDE, 2011 No. 15, p. 3 A weak solution to ( 4) is understood as such a function x ∈ H 1 0 (0, 1) that for all g ∈ H 1 0 (0, 1) the following relation holds: Lemma 1 We assume F1, F2, F3.For any fixed u ∈ L M functional J u is well defined and Gâteaux differentiable onto H 1 0 (0, 1).Moreover, weak solutions to (4) correspond to critical points of J u .

Proof. Let us fix any
. We further observe by inequality max that there exists a number Hence by the Mean Value Theorem, by integrability of t → F (t, 0) it follows by ( 5) that the integral dt is finite.By (6) we have also the integral (6).A direct calculation shows d dx J u (x u ) , g = 0 equals exactly (7).
We conclude this section with examples of nonlinearities satisfying our assumptions. Let , where g ∈ C 1 (R) has a bounded derivative and where s is an even number, and belongs to L 2 (0, 1).We remark that F 1 need not be convex on R and that t → (F 1 ) * (t, 0) is integrable.Indeed, for a.e.(fixed

Dependence on parameters for action functionals
In order to derive the results concerning the dependence on parameters for problem (4), we employ the following general principle.Let E be a Hilbert space with inner product •, • and with the induced norm • .Let C be a Banach space with norm • C .Let us consider a family of action functionals x → J (x, u), where x ∈ E and where u ∈ C is a parameter.
Theorem 2 Assume that E ∋ x → J (x, u) satisfies Palais-Smale condition, is weakly lower semicontinuous and bounded from below for any fixed u ∈ M, where M ⊂ C. Then x → J (x, u) has the argument of a minimum over E. Suppose further that there exists a constant α > 0 such that the set {(x, u) : ⊂ M be a weakly convergent sequence of parameters, where a weak limit lim n→∞ u n = u ∈ M. Let {x n } ∞ n=1 ⊂ E be the corresponding sequence of the arguments of minimum to E ∋ x → J (x, u n ).Then, there EJQTDE, 2011 No. 15, p. 5 we obtain that x is an argument of a minimum to x → J (x, u).
Proof.Let us fix u ∈ M. Since x → J (x, u) satisfies Palais-Smale condition, is weakly lower semicontinuous and bounded from below, it follows that J (•, u) has an argument of a minimum.
Let {u n } ∞ n=1 ⊂ M be a weakly convergent sequence of parameters with lim n→∞ u n = u.Now since the set {x : J (x, u) ≤ α} is bounded it follows that sequence {x n } ∞ n=1 ⊂ {x : J (x, u) ≤ α} of the arguments of a minimum to x → J (x, u n ) has a weakly convergent subsequence {x n i } ∞ i=1 ⊂ E. Let us denote x = lim i→∞ x n i , where x denotes the weak limit.
We will prove that x is an argument of a minimum to x → J (x, u).We see that there exists x 0 ∈ E such that J (x 0 , u) = inf y∈E J (y, u) and there are two possibilities: either J (x 0 , u) < J (x, u) or J (x 0 , u) = J (x, u).If we have J (x 0 , u) = J (x, u), then we have the assertion.Let us suppose that J (x 0 , u) < J (x, u), so there exists δ > 0 such that J (x, u) − J (x 0 , u) > δ > 0. (10) We investigate the inequality which is equivalent to (10).In view of (9) we see that the second and third term converge to 0. Finally, since Summarizing, we see that we have δ ≤ 0 in (11), which is a contradiction.EJQTDE, 2011 No. 15, p. 6 Theorem 3 Let u ∈ L M be arbitrarily fixed.Assume F1, F2, F3 .There exists x u ∈ H 1 0 (0, 1) such that J u (x u ) = inf x∈H 1 0 (0,1) J u (x) and Moreover, x u satisfies (4) for a.e.t ∈ [0, 1].
Proof.First we show that J u is weakly l.s.c. on H 1 0 (0, 1).Let us take any sequence {x n } ∞ n=1 ⊂ H 1 0 (0, 1) such that x n converges weakly in H 1 0 (0, 1) to x.Then {x n } ∞ n=1 contains by the Arzela-Ascoli Theorem a subsequence convergent uniformly and which we denote by Now by (8) we see that Since the remaining terms of J u are convex and defined on H 1 0 (0, 1), these are also weakly l.s.c. on H 1 0 (0, 1).Thus J u is weakly l.s.c. on H 1 0 (0, 1).
We observe that J u is coercive on H 1 0 (0, 1) in both cases.Indeed, in case F 1 is convex for any v ∈ R we get and further since t → F 1 x (t, 0) is integrable with square on [0, 1], the same follows for t → R (t) F 1 x (t, 0).Thus for any x ∈ H 1 0 (0, 1) EJQTDE, 2011 No. 15, p. 7 and by ( 6) In case t → (F 1 ) * (t, 0) is integrable we obtain by inequality Fenchel-Young inequality It follows that there exists x u ∈ H 1 0 (0, 1) such that J u (x u ) = inf x∈H 1 0 (0,1) J u (x) and obviously x u is a weak solution to (4).Applying the fundamental lemma of the calculus of variations we obtain that x u satisfies (4) for a.e.t ∈ [0, 1].
Next, by Lebesgue Dominated Convergence Theorem we see that lim and lim By the generalized Krasnosel'skij Theorem, see [3], and by (6) we see that strongly in L 2 (0, 1).Thus lim k→∞ u k = u weakly in L 2 (0, 1) provides that lim So by (19) we have lim kn→∞ J u kn (x kn ) − J u kn (x) = 0.
The same arguments lead to conclusion that Hence all the assumptions of Theorem 2 are satisfied.Thus x ∈ V u and so x necessarily satisfies (15).EJQTDE, 2011 No. 15, p. 10

Applications to optimal control
We now show the existence of an optimal process for an optimal control problem in which the dynamics is described by the Duffing equation, i.e. we will minimize the following action functional subject to (4) and where is measurable with respect to the first variable and continuous with respect to the two last variables and convex in u.Moreover, for any d > 0 there exists a function ψ d ∈ L 1 (0, 1) such that |f 0 (t, x, u)| ≤ ψ (t) a.e. on [0, 1] for all x ∈ [−d, d] and for all u ∈ M.
We define a set A consisting of pairs (x u , u) ∈ V u × L M on which we consider the existence of an optimal process to (20)-( 4); x u is a solution to (4) corresponding to u.We mention here that since the functions from L M are equibounded we get lim k→∞ u k = u weakly in L 2 (0, 1), up to a subsequence, for any sequence {u k } ∞ k=1 ⊂ L M .Moreover, any sequence {x k } ∞ k=1 , x k ∈ V u k or x k ∈ X, of solutions to (4) corresponding to such {u k } ∞ k=1 is necessarily bounded in H 1 0 (0, 1) as follows from the proof of Theorem Theorem 5 We assume f0, F1, F2, F3.There exists a pair (x, u) ∈ A such that J (x, u) = inf (xu,u)∈A J (x u , u).
Proof.Since any bounded sequence in H 1 0 (0, 1) has a uniformly convergent subsequence and by convexity of f 0 with respect to u we see that J is weakly l.s.c. on H 1 0 (0, 1) × L 2 (0, 1).Assumption f0 and remarks proceeding the formulation of the theorem provide that the functional J is bounded from below on A. Thus we may choose a minimizing sequence x k u , u k ∞ k=1 for a functional J such that u k ∞ k=1 is weakly convergent in L 2 (0, 1) to a certain u ∈ L M .Theorem 4 asserts that x k u ∞ k=1 converges, possibly up to a subsequence, strongly in H 1 0 (0, 1), weakly in H 1 0 (0, 1), strongly in C (0, 1) to a certain x solving (4) for u.Thus J (x, u) = lim inf k→∞ J x k u , u k ≥ J (x, u) ≥ inf 4. Thus there exists a d > 0 such that x k (t) ∈ [−d, d] for all k = 1, 2, ... and for a.e.t ∈ [0, 1].