Gradient flow approximation for stabili...

Motion planning for nonlinear control systems is one of the most important problems of mathematical control theory because of its theoretical challenges and many practical applications. Considerable efforts have been made to the solution of stabilization and motion planning problems for static target and obstacles. However, the complexity of control design increases significantly for problems related to dynamic environments, where the locations of targets and obstacles change with time.

The main goal of the project is to develop a general framework for stabilization and motion planning of nonholonomic systems governed by driftless control-affine systems. The main idea is to guarantee that the motion of the system steers along the approximated gradient flow of a certain time-varying potential function, for example, Lyapunov functions for stabilization and navigation functions for obstacle avoidance. Depending on the approximation method, two types of controls will be constructed: gradient-based and gradient-free. Gradient-based controls may explicitly depend on the derivatives of a potential function and can be used in situations where its analytical expression is known. In a variety of practical scenarios, however, an explicit analytical expression of the potential function is partially or completely unknown. For such cases, it is planned to construct gradient-free controls, which, nevertheless, ensure the gradient-like behavior of the system.