Optimal Control of a Bilinear System with a Quadratic Cost Functional

Various control systems in engineering problems are modelled with linear differential equations where a linear control is used. On the other hand, linear models are not capable of representing many systems where the control is applied in a multiplicative ways. These multiplicative controls yield bilinear systems (BLS). Products of state and control take part in BLS, which means that state and control are linear separately but not jointly. In this paper, optimal control of bilinear systems with a quadratic cost functional is studied. A distributed parameter system is considered and a bilinear control is applied to the system. The control problem is turned into a modal control problem by way of reduced order modelling. Performance index (cost functional) is defined as a measure of the dynamic response and a penalty term on control energy. Pontryagins maximum principle is used to obtain the optimal control function that leads to a nonlinear two-point boundary value problem. Optimal control and optimal trajectory of the system are determined by solving this two-point boundary value problem using steepest descent method. The programming is done in MATLAB platform. Numerical results are given in graphics for showing the effectiveness and applicability of the introduced bilinear control for a parabolic distributed parameter system.