We investigate the approximate controllability of a size- and space-structured population model, for which the control function acts on a subdomain and corresponds to the migration of individuals. We establish the main result via the unique continuation property of the adjoint system. The desired controller is the minimizer of an infinite-dimensional optimization problem.

In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini’s criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier−Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini’s criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.

We discuss several new results on nonnegative approximate controllability for theone-dimensional Heat equation governed by either multiplicative or nonnegative additivecontrol, acting within a proper subset of the space domain at every moment of time. Ourmethods allow us to link these two types of controls to some extend. The main resultsinclude approximate controllability properties both for the static and mobile controlsupports.

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

In this paper we prove a unique continuationresult for a cascade system of parabolic equations, in which the solution of the firstequation is (partially) used as a forcing term for the second equation. As aconsequence we prove the existence of ε-insensitizing controls for someparabolic equations when the control region and the observability region do not intersect.

We show that the set of nonnegative equilibrium-like states, namely, like $ (y_d, 0) $ of the semilinear vibrating string that can be reached from any non-zero initial state $ (y_0, y_1) \in H^1_0 (0,1) \times L^2 (0,1)$ , by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $ L^2 (0,1) \times \{0\} $ of $ L^2 (0,1) \times H^{-1} (0,1)$ . Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $ \; y \; $ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.

We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity and structure of f (x, t, u) in x and t and the choice of the space for controllability. We also show that our results are well-posed in terms of the “actual steering” of the system at hand, even in the case when it admits non-unique solutions.