Domain invariance for local solutions of semilinear evolution equations in Hilbert spaces

Piermarco Cannarsa (1) , Giuseppe da Prato (2) , Hélène Frankowska (3)

1 DIPMAT - Dipartimento di Matematica [Roma II]
2 SNS - Scuola Normale Superiore di Pisa
3 IMJ - Institut de Mathématiques de Jussieu

A closed set K of a Hilbert space H is said to be invariant under the evolution equation X (t) = AX(t) + f t, X(t) (t > 0) whenever all solutions starting from a point of K, at any time t0 0, remain in K as long as they exist. For a self-adjoint strictly dissipative operator A, perturbed by a (pos-sibly unbounded) nonlinear term f , we give necessary and sufficient conditions for the invariance of K, formulated in terms of A, f , and the distance function from K. Then, we also give sufficient conditions for the viability of K for the control system X (t) = AX(t) + f t, X(t), u(t) (t > 0, u(t) ∈ U). Finally, we apply the above theory to a bilinear control problem for the heat equation in a bounded domain of R N , where one is interested in keeping solutions in one fixed level set of a smooth integral functional.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Piermarco Cannarsa, Giuseppe da Prato, Hélène Frankowska. Domain invariance for local solutions of semilinear evolution equations in Hilbert spaces. Journal of the London Mathematical Society, 2020, ⟨10.1112/jlms.12320⟩. ⟨hal-02906910⟩