An Exact Penalty Method for Free Terminal Time Optimal Control Problem with Continuous Inequality Constraints

Abstract

In this paper, we consider a class of optimal control problems with free terminal time and continuous inequality constraints. First, the problem is approximated by representing the control function as a piecewise-constant function. Then the continuous inequality constraints are transformed into terminal equality constraints for an auxiliary differential system. After these two steps, we transform the constrained optimization problem into a penalized problem with only box constraints on the decision variables using a novel exact penalty function. This penalized problem is then solved by a gradient-based optimization technique. Theoretical analysis proves that this penalty function has continuous derivatives, and for a sufficiently large and finite penalty parameter, its local minimizer is feasible in the sense that the continuous inequality constraints are satisfied. Furthermore, this local minimizer is also the local minimizer of the constrained problem. Numerical simulations on the range maximiCommunicated by George Leitmann. J Optim Theory Appl (2012) 154:30-53 31 zation for a hypersonic vehicle reentering the atmosphere subject to a heating constraint demonstrate the effectiveness of our method.

An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

“…where is defined in (27), is a homeomorphism. Moreover, this mapping is differentiable on int.C/n ¹0º.…”

“…Determine the gauge function G C i of each C i defined in (13) and constitute the generalized saturation functions according to (27)- (28). For example, if u must belong to OEa; b, then pick: .…”

“…There exist also recent works in the field of exact penalty methods for various types of optimal control problem [24][25][26][27][28][29]. These methods are of particular interest because each solution of the sequence of optimal control problem is easily computed using classical stationarity conditions of the solution.…”

A computational algorithm for a class of non-smooth optimal control problems arising in aquaculture operations

“…Following their lead, we also ignore the feeding rate f (t) and consider only the harvesting fractions and the corresponding harvesting times as decision variables. However, note that the computational approach in this paper can be easily extended to also optimize the feeding rate f (t) using the control parameterization technique described in [1,4,6].…”

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