Assessing optimality and robustness for the control of dynamical systems

This work presents a general framework for assessing the quality and robustness of control over a deterministic system described by a state vector [Formula Presented] under external manipulation via a control vector [Formula Presented] The control process is expressed in terms of a cost functional, including the physical objective, penalties, and constraints. The notions of optimality and robustness are expressed in terms of the sign and the magnitude of the cost functional curvature with respect to the controls. Both issues may be assessed from the eigenvalues of the stability operator S whose kernel [Formula Presented] is determined by [Formula Presented] for [Formula Presented] [Formula Presented] where [Formula Presented] and [Formula Presented] are the initial and final times of the control interval. The overbar denotes the constraint that the control satisfies the optimization conditions from minimizing the cost functional. The eigenvalues σ of S satisfying [Formula Presented] assure local optimality of a control solution, with [Formula Presented] being the critical value separating optimal solutions from false solutions (i.e., those with negative second variational curvature of the cost functional). In turn, the maximally robust control solutions with the least sensitivity to field errors also correspond to [Formula Presented] Thus, sufficiently high sensitivity of the field at one time t to the field at another time τ (i.e., [Formula Presented] will lead to a loss of local optimality. A simple illustrative example is given from a linear dynamical system, and a bound for the eigenvalue spectrum of the stability operator is presented. The bound is employed to qualitatively analyze control optimality and robustness behavior. A second example of a nonlinear quartic anharmonic oscillator is also presented for stability and robustness analysis. In this case it is proved that the control system kernel is negative definite, implying full stability but only marginal robustness.