Hybrid Optimal Control

Optimal control of Hybrid Ordinary Differential Systems


[Presentation] [Papers] [Code] [Contacts]

Jean-Guillaume Dumas


This project consists in providing generic algorithms with both symbolic and numerical modules to solve nonlinear optimal control problems.

Jean-Guillaume Dumas

We consider a general nonlinear dynamical system which state is described by the solution of the following ODE:

We want to present a generic algorithm for controlling the system (1) from an initial state X0 at time t=0 to a final state Xf at unspecified time tf using the admissible control functions u that take values in a convex and compact polyhedral set Umof Rm, in such a way that:

is minimized. The polytope Um is defined as the convex hull of its vertices: 

Jean-Guillaume Dumas


We first consider general linear systems. The system (1) is replaced by the following one: X'(t)=AX(t) + Bu(t). We then provide a generic algorithm with both symbolic and numerical modules. Especially we propose new algorithm under-approximating  the controllable domain in view of its analytical resolution in the context of singular subarcs. New efficient methods computing a block Kalman canonical decomposition and the optimal solutions are also presented.

Jean-Guillaume Dumas


Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given an optimal sequence of states, we  are then able to traverse the automaton till the target, locally insuring the optimality.

Jean-Guillaume Dumas


For further informations about this work, have a look at:


Jean-Guillaume Dumas

[Source Code]

You can also download our source code:

                Available in this library:
        (Full description and user's guide soon available)

To run it, you need:

Jean-Guillaume Dumas


Laboratoire de Modélisation et Calcul
B.P. 53 -- 51, av. des Mathématiques,
38041 Grenoble, France.
Jean-Guillaume DUMAS
Laboratoire de Modélisation et Calcul
B.P. 53 -- 51, av. des Mathématiques,
38041 Grenoble, France

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