Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green's function approach

Abstract

A systematic method for establishment of exact and approximate controllability of dynamic systems with nonlinear state constraints within infinite time is developed. Using the recently developed Green's function approach, we derive necessary and sufficient conditions for exact controllability, as well as sufficient conditions for approximate controllability. Conditions for null‐controllability and lack of controllability are established as well. Several possibilities of deriving explicit form for resolving controls are described. Possibilities of heuristic determination of resolving controls is studied in details. The advantages and drawbacks of the method are revealed on specific examples of nonlinear wave and diffusion equations in unbounded domains.

The Green’s Type Matrices for Consumption Reduction in a Heterogeneous Population Model of TCLs with Diffusion

“…Since the mathematical model is linear in (u, v), for our purpose, we involve the Green's function approach developed in [13]. Substituting (14) into (15), we make the dependence R = R Θ sp , σ explicit.…”

“…], we choose the system parameters according to Table 1. Involving the numerical scheme for computing the elements of the matrix of Green's type G ij , i, j = 1, 2, developed in [14], we compute (u, v) according to (13). Then, κ is computed using (14).…”

Controlling Power Consumption in a Heterogeneous Population Model of TCLs with Diffusion: The Green’s Function Approach

“…Since the mathematical model is linear in (u, v), for our purpose, we involve the Green's function approach developed in [14]. Substituting (14) into (15), we make the dependence R = R Θ sp , σ explicit.…”

Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

“…Nonlinear wave equations like (14) arise, e.g., in biology [17], in many areas of physics, mechanics, and engineering, describing, as a rule, nonlinear vibrations in solids or fluids [18]. In particular, they describe vibrations of a pendulum, vibrations of nonlinear elastic rods, nonlinear electromagnetic oscillations, nonlinear gravitational waves, etc.…”

“…Note that this approach is applicable as long as nonlinear Green's equation (4) is resolvable under the corresponding Cauchy conditions. Note also that the idea of [13,14] can be applied on the results of this paper, in order to consider control problems for new nonlinear dynamic systems described by the "oscillating" equations (1) and related partial differential equations.…”