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

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 (possibly 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 RN, where one is interested in keeping solutions in one fixed level set of a smooth integral functional.