Min-max optimal control of systems approximated by finite-dimensional models: Nonquadratic cost functional

An open-loop control problem with nonquadratic performance criterion for a dynamical system, described by an abstract linear equation of evolution and approximated by a finite-dimensional model, is solved. A min-max approach is taken: the value of the cost functional in theworst-case output error between the system and the model, under the assumption that the norm of this output error is estimated by the norm of the input, is minimized.The form of the cost functional reproduces itself under maximization, so that the min-max control problem, when only thedistance between the model and the system is given, has the same features and proprieties of the control problem when the system is thorougly known.Existence and uniqueness theorems for the optimal control are proven, using the spectral proprieties of the model transfer function; and, in the case of a time-invariant model, the min-max control computation is reduced to the solution of a constant-coefficients Sturm-Liouville problem followed by the search for the zeros of a very simple numerical function.This work was made within the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche.