A Mathematical Framework for Controllability of Single-Agent Systems Modelled as a Point-Particle Governed by Linear Dynamics in Continuous-Time

This report serves as a pedagogical exposition to the controllability of single-agent systems. The single-agent systems are ubiquitous both in the natural and technological domains. A single-agent system modelled as a point-particle within a state space $\mathbb{R}^n$ whose dynamics is governed by a linear continuous-time ordinary differential equation of $m^{\text{th}}$-order is considered in the report. The work presents a functional operator-theoretic framework for analysing the controllability of such single-agent systems. In particular, controllability is characterised through the surjectivity of the control operator, the injectivity of the adjoint of the control operator, and the positive definiteness of the controllability Gramian matrix. In the time-invariant case, these conditions reduce to the Kalman's rank criterion. we realise that, if the system is controllable over a time domain, then it is always controllable over a sup-time domain. The vice versa is also true in case of time-invariant systems. A brief review of the trajectory controllability, which is a stronger notion of (state) controllability, is given. The necessary and sufficient condition for the trajectory controllability is proposed. Unlike in the case of (state) controllability, here it is realised that if the system is trajectory controllable over a time domain, then it is always trajectory controllable over a sub-time domain. The vice versa is also true in case of time-invariant systems. The examples included in the document show that the theory applies across a wide range of models. We consider the radioactive decay system, the water-tank problem, and the resistor–capacitor circuit in electrical network, and illustrates how simple linear models can be controlled and stabilised.