By Alexander Y. Khapalov
The target of this monograph is to handle the difficulty of the worldwide controllability of partial differential equations within the context of multiplicative (or bilinear) controls, which input the version equations as coefficients. The mathematical types we study contain the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and matched hybrid nonlinear dispensed parameter platforms modeling the swimming phenomenon. The e-book deals a brand new, fine quality and intrinsically nonlinear technique to procedure the aforementioned hugely nonlinear controllability problems.
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Extra info for Controllability of partial differential equations governed by multiplicative controls
Based on an asymptotic technique allowing us to separate and combine the impacts generated by the abovementioned two types of controls, we manage to establish the global approximate controllability for such superlinear pde in the classical sense. Chapter 3 Multiplicative Controllability of the Semilinear Parabolic Equation: A Qualitative Approach Abstract In this chapter we establish the global non-negative approximate controllability property for a rather general semilinear heat equation with superlinear term, governed in a bounded domain Ω ⊂ Rn by a multiplicative control in the reaction term like vu(x,t), where v is the control.
2 Main Results Exact null-controllability. Our main results here are as follows. 1. e. 2) for some positive constant ν0 > 0. 1) vanishes at time T : u(·, T ) = 0. a. (x,t) ∈ Q∞ . e. in Q∞ . 6)). 4 below. , ). 1. 6). 1. 1). Our next result deals with the case when n = 1 and α ∈ L2 (QT ) vanishing outside of the given strict subdomain of Ω .
1) but now in several space dimensions with the terms f which can again be superlinear at infinity but now they do not have to be superlinear near the origin. Contrary to the above, the main result of Chapter 3 always requires that at least three “large” static bilinear controls (whose magnitudes increase as the precision of steering increases) be applied subsequently for very short times. Unlike the method of the present section, based on the use of the dynamics imposed by the diffusion-reaction term like yxx + α (x)y, the method of Chapter 3 focuses on the “suppression” of the effect of diffusion.