Setting of Controllability ProblemIn this chapter we discuss multiplicative controllability of hyperbolic equations along the approach due to J.M. Ball, J.E. Marsden, and M. Slemrod. We describe the main ideas of this approach and the principal results relevant to the global reachability properties of these equations. For the complete account of all details we refer the reader to the original paper [8].Chapter 9 is organized as follows. In section 9.1 a class of abstract evolution equations, governed in a Banach space X by multiplicative time-dependent controls, is introduced. However, it turns out that the local controllability of these equations (i.e., anywhere within some neighborhood of a drifting trajectory) is out of question when X is infinite dimensional. In section 9.2 we distinguish a more specific subclass within the aforementioned class of equations -the abstract hyperbolic equations, and also remind the reader the concept of Riesz bases. In section 9.3 we show how the global approximate reachability of the latter equations can be established under a number of additional assumptions. In particular, it is assumed that the eigenvalues of the linear part of the equation at hand must all be integers that are multiplied by the same number and that all modes in the initial datum must be present (see Assumptions 1-4 below). In section 9.4 this abstract theory is applied to the 1-D wave and rod equations. In section 9.5 we discuss two results on local controllability for these equations due to K. Beauchard.

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We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state u * ∈ H 1 0 (0, 1), with as many changes of sign in the same order as the given initial data u0 ∈ H 1 0 (0, 1), can be approximately reached in the L 2 (0, 1)-norm at some time T > 0. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems.

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Abstract.We study the global approximate controllability of the one dimensional semilinear convection-diffusion-reaction equation governed in a bounded domain via the coefficient (bilinear control) in the additive reaction term. Clearly, even in the linear case, due to the maximum principle, such system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in L 2 (0, 1) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static (x-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.Mathematics Subject Classification. 93, 35.