Every Continuous Nonlinear Control System Can be Obtained by Parametric Convex Programming

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

“…Conversely, given a continuous PWA function defined over a polyhedral partition, the question is whether there exists an optimization problem such that its optimal solution is equivalent to the given PWA function. The answer is shown in [10] to be affirmative, although the numerical construction of such an optimization problem is still open. A possible candidate for this optimization problem may be characterized by a linear/quadratic cost function and a set of linear constraints.…”

“…In the context of inverse optimality, the idea of using this optimal cost function as an auxiliary variable is proposed in [10] and is recalled here. For ease of presentation, let µ(x) denote the controller defined over a convex set Ω to be recovered and g 0 (x) represent a strictly convex function over Ω.…”

“…It is worth stressing that a convex lifting used in this paper is not strictly convex, but convex, as there does not exist a strictly convex, piecewise affine function. Moreover, the recovered optimization problem is shown to have the unique optimal solution which was not treated in [10].…”

“…This inverse optimality problem has been investigated for some years and has resulted in interesting results for the general nonlinear continuous functions [10] and recently for continuous PWA functions [11,12].…”

“…The authors in [10] proved that every continuous feedback law can be obtained by PCP. This is an insightful mathematical result; however, it remains purely theoretical; neither a constructive procedure nor a qualitative interpretation of the dimension of the optimization arguments is provided.…”