Korobov’s controllability function method via orthogonal polynomials on [0,∞)

Given a controllable system described by ordinary or partial differential equations and an initial state, the problem of finding a set of bounded positional controls that transfer the initial state to another state, not necessarily an equilibrium point, in finite time is called the synthesis problem. In the present work, we consider a family of Brunovsky systems of dimension n. A family of bounded positional controls un(x) is developed to stabilize a given Brunovsky system in finite time. We employ orthogonal polynomials associated with a function distribution σ(τ, θ) defined for τ ∈ [0, +∞) and parameter θ > 0. The parameter θ is interpreted as a Korobov’s controllability function, θ = θ(x), which serves as a Lyapunovtype function. Utilizing θ(x), we construct the positional control un(x) = un(x, θ(x)). Our analysis is based on the foundational work “A general approach to the solution of the bounded control synthesis problem in a controllability problem”. Matematicheskii Sbornik, 151(4), 582–606 (1979) by Korobov, V. I, in which the controllability function method was proposed. This method has been applied to solve bounded finite-time stabilization problems in various control scenarios, such as the control of the wave equation, optimal control with mixed cost functions, and other applications. For the construction of the mentioned positional controls, we employ a member of a family of orthogonal polynomials on [0,∞). For orthogonal polynomials, we refer to “Orthogonal Polynomials”. American Mathematical Society, Providence, (1975) by G. Szego. We also rely on the work “On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials”. Linear Algebra and its Applications, 476, 56–84 (2015) by Choque Rivero, A. E. The results in the present work extend and develop the findings presented in the conference paper “Bounded finite-time stabilizing controls via orthogonal polynomials”. 2018 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC), Ixtapa, Mexico. –2018 by Choque-Rivero A. E., Orozco B. d. J. G.