Quasiconvex relaxation of multidimensional control problems with integrandsf(t,ξ,v)

Abstract

Abstract. We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f (t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.Mathematics Subject Classification. 26B05, 26B25, 49J20, 49J45, 68U10.

On the integral representation of relaxed functionals with convex bounded constraints

“…In the vectorial case, and in the presence of some singular behaviors of the stored energy functions in nonlinear elasticity, we can find some relaxation results where the integrands can take the value +∞, see [2,3,5]. Moreover, recently, in connection with relaxation problems in optimal control, Wagner [16] studies the relaxation of integral functional with the assumption that f is continuous finite on C, and infinite elsewhere.…”

Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems

“…In Section 4, we compare the relaxation approaches for (P) mentioned in Section 1.1 above. Concerning the first approach with the replacement of the integrand, we recall the result from [41] about the problem (P) (qc) . Pursuing the second approach, we formulate a relaxed problem ( P) (qc) with generalized controls and prove then the identity of the minimal values of (P) (qc) and ( P) (qc) .…”

“…(See[41, p. 309, Theorem 1.3].) Consider the problem (P), assuming that m 2, n 1, K ⊂ R nm is an arbitrary convex body with o ∈ int(K), and the integrand f belongs to F K .…”

“…(See[41, p. 320, Lemma 3.6].) Let functions f ∈ F K and u ∈ L ∞ (Θ, R nm ) with u(t) ∈ K (∀)t ∈ Θ be given.…”

A note on locking materials and gradient polyconvexity

“…In Section 3, we study relaxation under the constraint L(∇y) ≤ 0 with L as in (1.6) and elastic boundary conditions (see (3.1)) by means of gradient Young measures. Indeed, while the vast majority of relaxation techniques available in the literature concern only energies that take finite values, the only relaxation result under the constraint based on (1.6), to the best of the authors' knowledge, is due to Wagner [61] (see also [59,60]), who characterized the relaxed energy by means of an infimum formula. We also refer to [14,22] for relaxation results of unbounded functionals with scalar-valued competing maps and to [15,25] for homogenization problems for unbounded functionals.…”