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Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction. (English) Zbl 0880.70020
Summary: We establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure, and thus it leads rather directly to the desired conclusions for such systems. We apply our method to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). The main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation.
MSC:
70Q05Control of mechanical systems (general mechanics)
70F20Holonomic systems (particle dynamics)
70F25Nonholonomic systems (particle dynamics)
49J15Optimal control problems with ODE (existence)