New smoothing techniques for solving bang–bang optimal control problems—numerical results and statistical interpretation

Abstract

SUMMARYIn this paper, we investigate the solution of bang-bang optimal control problems by shooting methods. We will show how modifying the performance index by a term depending on a small parameter e yields more regular controls and shooting functions. A continuation procedure on e will lead us to a good approximation of the initial solution. Then, a statistical interpretation of the method is given, providing us with a general framework for building new regular controls. Finally, two numerical examples are solved illustrating the interest of our method.

Multi-step optimization strategy for fuel-optimal orbital transfer of low-thrust spacecraft

“…Note that, although derived in a different way, the exponential representation can be already found in Bertrand and Epenoy (2002). Differently from Bertrand and Epenoy (2002), in the present approach the continuation method is applied only to the control representation, while the objective function remains unchanged during the iterations. By heuristically increasing the value of p, the discontinuous behaviour of the control is gradually approximated.…”

“…This example is presented by Bertrand and Epenoy (2002), and it is also analyzed by Jiang et al (2012). Both works make use of an indirect approach, although the continuation method adopted is applied to the objective function rather than on the control representation.…”

“…This includes assigning the switching structure a priori (La Mantia and Casalino 2006), using the model of fuel consumption rate proportional to the quadratic of thrust magnitude (Vadali 2001), and reducing the dimension of the problem considered (Mengali and Quarta 2007). Bertrand and Epenoy (2002) also proposed the use of smoothing techniques with a continuation method to limit numerical issues and solve the interplanetary transfer problem accurately.…”

Minimizing the fuel consumption of a vehicle from the Shell Eco-marathon: a numerical study

“…where the control function U now takes values in the interval [0,1]. This "continuous relaxation" means that we are not restricted to on/off engine switches anymore: we are also allowed to accelerate more or less.…”

“…This study was prompted by Yves Mégret, the leader of the Toulouse Ingénierie Mécanique team. The author is also thankful to Bonnans [2] and Epenoy [1] for interesting discussions about logarithmic continuation, Jean-Baptiste Hiriart-Urruty for his encouragement, Marcel Mongeau and Aude Rondepierre for their help, Sager [9] for a test of his code, and Emmanuel Trélat for more discussions about the continuation approach (quadratic versus logarithmic).…”