Approximate Controllability for Degenerate Heat Equation with Bilinear Control
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Approximate Controllability for Degenerate Heat Equation with Bilinear Control∗ LI Lingfei · GAO Hang
DOI: 10.1007/s11424-020-9082-3 Received: 10 March 2019 / Revised: 16 November 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract This paper investigates the nonnegative approximate controllability for the one-dimensional degenerate heat equation governed by bilinear control. Both non-controllability and approximate controllability are studied for the system. If the control is restricted to act on a fixed domain, it is not controllable. If the control is allowed to mobile, it is approximately controllable. Keywords
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Approximate controllability, bilinear control, degenerate heat equation.
Introduction and Main Results
Mathematical cybernetics refers to apply mathematical methods to study control problems. It mainly consists of structural analysis and design of control systems. At present, mathematical cybernetics mainly discusses three types of systems, namely, finite-dimensional systems, distributed parameter systems and random systems. In recent decades, the control theory of distributed parameter systems and random systems has been greatly developed by many scholars (see, for instance, [1–9]). The bilinear systems are usually used to describe reaction-diffusion convection processes such as chemical reactions by catalysts, biological chain reactions. The dependence of the state function with respect to the control function is highly nonlinear. The traditional duality argument is not suitable for the bilinear problems. In the last decades, the controllability of bilinear systems has been extensively studied. In the pioneering work [9], the global approximate controllability of the rod and wave equation was established by using the nonharmonic Fourier series approach. By exploring the idea of [9], the authors in [10] discussed the simultaneous LI Lingfei School of Science, Northeast Electric Power University, Jilin 132012, China. Email: [email protected]. GAO Hang (Corresponding author) School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China. Email: [email protected]. ∗ This research was supported by the National Natural Science Foundation of China under Grant Nos. 11771074, 11871142 and the PhD Research Start-up Fund of Northeast Electric Power University under Grant No. BSJXM2019113. This paper was recommended for publication by Editor LIU Yungang.
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LI LINGFEI · GAO HANG
control of the rod and Schr¨ odinger equation. In [11, 12], Khapalov achieved the nonnegative approximate controllability of the parabolic system with semilinear/superlinear term by acting static bilinear controls subsequently in time. Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes has been investigated in [13, 14]. The exact controllability of bilinear systems can be found in [15–19]. Several applied models can be described as degenerate parabolic equations, for example, the Budyko-Sellers model in climatology
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