DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019092 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 5, May 2019 pp. 2219–2235 ON NUMERICAL METHODS FOR SINGULAR OPTIMAL CONTROL PROBLEMS: AN APPLICATION TO AN AUV PROBLEM Z. Foroozandeh * and Maria do Ros´ ario de Pinho Faculdade de Engenharia Universidade do Porto, DEEC, Porto, Portugal M. Shamsi Department of Applied Mathematics, Faculty of Mathematics and Computer Science Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran Abstract. We discuss and compare numerical methods to solve singular op- timal control problems by the direct method. Our discussion is illustrated by an Autonomous Underwater Vehicle (AUV) problem with state constraints. For this problem, we test four different approaches to solve numerically our problem via the direct method. After discretizing the optimal control problem we solve the resulting optimization problem with (i) A Mathematical Pro- gramming Language (AMPL), (ii) the Imperial College London Optimal Con- trol Software (ICLOCS), (iii) the Gauss Pseudospectral Optimization Software (GPOPS) as well as with (iv) a new algorithm based on mixed-binary non- linear programming reported in [7]. This algorithm consists on converting the optimal control problem to a Mixed Binary Optimal Control (MBOC) problem which is then transcribed to a mixed binary non-linear programming problem (MBNLP) problem using Legendre-Radau pseudospectral method. Our case study shows that, in contrast with the first three approaches we test (all relying on IPOPT or other numerical optimization software packages like KNITRO), the MBOC approach detects the structure of the AUV’s problem without a priori information of optimal control and computes the switching times accu- rately. 1. Introduction. This paper focuses on numerical solution of singular optimal control problems (SOCPs) via the direct method. We test and compare four differ- ent approaches applied to a problem for Autonomous Underwater Vehicle (AUV’s). Our illustrative problem is a singular optimal control problem (SOCP) involving a simplified model for the path planning of an AUV’s in a horizontal plane treated previously in [16] and [5]. In SOCPs the control appears linearly in the dynamics and cost and it has lower and upper bounds. It is well known that the optimal control for those problems is a concatenation of bang-bang or bang-singular type control [14, 13, 8]. The main difficulty when solving singular optimal control problems numerically lies in the determination of the switching structure of the optimal control, i.e., the sequence 2010 Mathematics Subject Classification. Primary: 49J30, 65K05; Secondary: 49M37. Key words and phrases. Singular optimal control problem, switching points, AUV’s problem, Legendre-Gauss-Radau pseudospectral method, mixed-binary non-linear programming, mixed- binary optimal control, ICLOCS, GPOPS, Implicit Euler Method. * Corresponding author: Z. Foroozandeh. 2219