Theory and method of the phase portrait of dynamic systems with control (Q1083685)

The concept of phase portrait of controlled dynamical systems is presented. The dynamics of the system is described by an autonomous differential inclusion \(\dot q\in F(q)\), \(q\in M\subset R^ n\), \(F(q)\subset T_ qM\) where M is a smooth differentiable manifold and \(T_ qM\) denotes the tangent space of M at the point q. It is assumed that F(q) is a non-empty, compact and convex set in \(R^ n\). The fundamental role is played by the cones K(q) of admissible directions of the velocities at every point \(q\in M\); K(q) is the conical hull of F(q). Some different types \(a^ m_ r\) (r\(\leq m\leq n)\) of those cones are defined by algebraic conditions and next in terms of the Hamiltonian; there is a finite number of such types of cones depending only on n. Next, the state space M is partitioned into disjoint subsets with the same type of cone at every point of such a subset. The phase portrait contains: the integral funnel at every point \(q\in M\), its boundary, singular manifolds regions and manifolds of local or complete controllability (in particular manifolds of unconstrained motions), separating surfaces etc. At the end the phase portrait for a two- dimensional system is presented.