Controllability of partial differential equations governed by Alexander Y. Khapalov

By Alexander Y. Khapalov

The aim of this monograph is to deal with the problem 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 comprise the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and paired hybrid nonlinear dispensed parameter platforms modeling the swimming phenomenon. The booklet deals a brand new, top quality and intrinsically nonlinear method to technique the aforementioned hugely nonlinear controllability problems.

Show description

Read or Download Controllability of partial differential equations governed by multiplicative controls PDF

Similar linear programming books

Linear Programming and its Applications

Within the pages of this article readers will locate not anything lower than a unified therapy of linear programming. with out sacrificing mathematical rigor, the most emphasis of the e-book is on types and functions. crucial periods of difficulties are surveyed and awarded via mathematical formulations, by means of answer tools and a dialogue of quite a few "what-if" eventualities.

Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems (Classics in Applied Mathematics, 37)

This article makes an attempt to survey the middle matters in optimization and mathematical economics: linear and nonlinear programming, setting apart aircraft theorems, fixed-point theorems, and a few in their applications.

This textual content covers basically topics good: linear programming and fixed-point theorems. The sections on linear programming are founded round deriving equipment in line with the simplex set of rules in addition to a number of the usual LP difficulties, reminiscent of community flows and transportation challenge. I by no means had time to learn the part at the fixed-point theorems, yet i feel it can turn out to be precious to analyze economists who paintings in microeconomic concept. This part offers 4 varied proofs of Brouwer fixed-point theorem, an explanation of Kakutani's Fixed-Point Theorem, and concludes with an explanation of Nash's Theorem for n-person video games.

Unfortunately, an important math instruments in use by means of economists this present day, nonlinear programming and comparative statics, are slightly pointed out. this article has precisely one 15-page bankruptcy on nonlinear programming. This bankruptcy derives the Kuhn-Tucker stipulations yet says not anything concerning the moment order stipulations or comparative statics results.

Most most probably, the unusual choice and assurance of subject matters (linear programming takes greater than 1/2 the textual content) easily displays the truth that the unique variation got here out in 1980 and likewise that the writer is de facto an utilized mathematician, now not an economist. this article is worthy a glance if you want to appreciate fixed-point theorems or how the simplex set of rules works and its functions. glance in different places for nonlinear programming or newer advancements in linear programming.

Planning and Scheduling in Manufacturing and Services

This publication specializes in making plans and scheduling functions. making plans and scheduling are sorts of decision-making that play a huge function in such a lot production and prone industries. The making plans and scheduling features in an organization ordinarily use analytical strategies and heuristic the right way to allocate its constrained assets to the actions that experience to be performed.

Optimization with PDE Constraints

This ebook offers a contemporary advent of pde limited optimization. It offers an actual useful analytic remedy through optimality stipulations and a state of the art, non-smooth algorithmical framework. in addition, new structure-exploiting discrete ideas and big scale, essentially proper functions are provided.

Additional info for Controllability of partial differential equations governed by multiplicative controls

Sample text

4 there). In this chapter we focus primarily on the global exact null-controllability problem, which requires a completely different method. 1) is globally exactly null-controllable in H if it can be steered in H from any initial state to the zero-state exactly. Y. 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. , [97]). 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 .

49], h(x,t1 ) > 0 in the interior of Ω , h(x,t1 ) |∂ Ω = 0. 3: Nonnegative Controllability 45 Step 2. Consider any t2 > t1 . On the interval (t1 ,t2 ) we apply a positive constant control v(x) = v (its value will be chosen later). 38) Ω where λk (λk → −∞ as k → ∞) and ωk (x) ( ωk L2 (Ω ) = 1), k = 1, . . are respectively the eigenvalues and eigenfunctions associated with the spectral problem Δ ω = λ ω , ω |∂ Ω = 0 in H01 (Ω ). Consider any number γ > 1 (its value will be chosen more precisely a little bit later) and select a constant (in t and x) control v > 0 such that ev(t2 −t1 ) = γ , namely, v = ln γ .

19)) will be ud (x) = ω1 (x). 4)), in place of the target state. 40) for some t ∗ > t∗ (where λk ’s are the eigenvalues associated with α∗ , λ1 = 0). 5) with α = α∗ . 42) where ρ > 0 is some (fixed) constant. Since λ1 = 0, a < 0 and α (x) = α∗ (x) + a < 0, x ∈ [0, 1]. 24) applies on the interval (t∗ ,t ∗ ): u(·,t ∗ ) − s1+ξ ud L2 (0,1) u(·,t ∗ ) − y(·,t ∗ ) ≤ L2 (0,1) r1 5 5 + y(·,t ∗ ) − s1+ξ ud ≤ C(t ∗ − t∗ )max{ 6 (1− 5 ), 6 (1− + Csξ λ2 /a s1+ξ ud 3r2 5 )} L2 (0,1) smin{r1 ,r2 } L2 (0,1) = o(s1+ξ ) as s → 0+ (we remind the reader that C denotes a generic positive constant).

Download PDF sample

Rated 4.76 of 5 – based on 25 votes