A common theme in all the above areas is designing a dynamical system to accomplish desired objectives, possibly in some predefined optimal way. Since control theory advances the idea of suitably modifying the behavior of a dynamical system, this paper explores the role of control theory in designing efficient algorithms (or dynamical systems) related to problems surrounding the optimization framework, including constrained optimization, optimization-based control, and parameter estimation. This amalgamation of control theory with the above-mentioned areas has been made possible by the recently introduced paradigm of Passivity and Immersion (P&I) based control. The generality and working of P&I, as compared to the existing approaches in control theory, are best introduced through the example presented below.

Gradient Descent (GD) is a ubiquitous algorithm for finding the optimal solution to an optimization problem. For reduced computational complexity, the optimal solution x* of the optimization problem must be attained in a minimum number of iterations. For this objective, the paper proposes a genesis of an accelerated gradient algorithm through the controlled dynamical system perspective. The objective of optimally reaching the optimal solution x* where the gradient of  ∇f(x*) is zero with a given initial condition x(0) is achieved through control.