Masamu Institute for Research, Training, and Advanced Studies

Abstract

Controllability is a fundamental concept in control theory concerned with the ability to steer a dynamical system toward a desired target state using admissible controls. We begin with the classical controllability theory of finite-dimensional systems, highlighting the Kalman rank condition and its role in characterizing exact controllability. We then move to infinite-dimensional systems, where the dynamics are typically governed by partial differential equations and exact controllability becomes considerably more restrictive, making approximate controllability a more suitable framework. In this talk, we study the approximate boundary controllability of infinite-dimensional systems described by partial functional integrodifferential equations incorporating nonlinearities, finite delays, and memory effects. Using Grimmer’s resolvent operator theory and fixed-point techniques, we establish approximate controllability under suitable assumptions on the associated linear system and nonlinear terms. The framework is further extended to stochastic systems driven by both Wiener and Rosenblatt processes, allowing for Gaussian and non-Gaussian long-range dependent perturbations. An illustrative heat equation with memory and boundary control demonstrates the applicability of the results. The talk concludes with perspectives on the control of nonlinear, stochastic, and memory-dependent infinite-dimensional systems.