Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

Abstract

In a separable Hilbert space X, we study the controlled evolution equation $$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0, \end{aligned}$$ u ′ ( t ) + A u ( t ) + p ( t ) B u ( t ) = 0 , where $$A\ge -\sigma I$$ A ≥ - σ I ($$\sigma \ge 0$$ σ ≥ 0 ) is a self-adjoint linear operator, B is a bounded linear operator on X, and $$p\in L^2_{loc}(0,+\infty )$$ p ∈ L loc 2 ( 0 , + ∞ ) is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the jth eigensolution for any $$j\ge 1$$ j ≥ 1 . We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.

Classical Solutions of a Mixed Problem with the Zaremba Boundary Condition for a Mildly Quasilinear Wave Equation

“…Definition 2.5. Following [43,44,45], we call a function u representable in the form (2.7), (2.8), (2.14), (2.10), where γ : [0, ∞) ∋ t → γ(t; ϕ(0) − µ 1 (0)) ∈ R is a function satisfying the condition (2.13), a mild solution of the problem (1.1) -(1.4).…”

Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation

“…In a series of recent papers (see [3,4,16]), we have studied stabilization and exact controllability to eigensolutions for evolution equations of the form u ′ (t ) + Au(t ) + p(t )Bu(t ) = 0, t ∈ (0, T )…”

“…The scalar-input bilinear controllability problem has been addressed by several authors, starting with the negative result by Ball, Marsden, Slemrod [7]. Controllability issues are interesting also in the hyperbolic or diffusive context, where several results are now available to describe the reachable set of specific partial differential equations in 1−D, such as the Schrödinger equation [5,8,10] and the classical [5,9] and degenerate [13] wave equation. The above problem enters in the so-called class of nonlinear control problems.…”

“…Most of the above results devoted to scalar-input bilinear controllability issues have in common the fact that they address controllability properties of (1) near the ground state solution ψ 1 (t ) = e −λ 1 t ϕ 1 (see [9,10,13]) or, more in general, to eigensolutions ψ j (t ) = e −λ j t ϕ j (see [3,4]), that are solutions of the free dynamics (p = 0) associated to (1), namely u ′ (t ) + Au(t ) = 0, with initial condition u 0 = ϕ j , where we denote by {λ k } k ∈ N * the eigenvalues of A, and by {ϕ k } k ∈ N * the associated eigenfunctions.…”

“…the control cost. Furthermore, we have shown that sufficient conditions for {A, B } to be j -null controllable are a gap condition of the eigenvalues of A and a rank condition on B : we assume B to spread ϕ j in all directions (see [4,Theorem 1.2]). When, for instance, X = L 2 (0, 1), B could be taken as a multiplicative operator, i.e., (B ϕ)(x) = µ(x)ϕ(x),…”