Time-optimal Control Of System Research Articles

Time Optimal Control of System Governed by a Fractional Stochastic Partial Differential Inclusion with Clarke Subdifferential

This paper investigates the time optimal control problems to a new class of fractional non-instantaneous impulsive stochastic partial differential inclusions with Clarke subdifferential in Hilbert spaces. Firstly, using the fractional calculus, properties of fractional resolvent operators and a fixed-point theorem, the existence of mild solutions for these systems is presented. Secondly, the existence of time optimal control of system governed by fractional stochastic control inclusions with Clarke subdifferential and non-instantaneous impulses is also obtained. Finally, an example is given to illustrate our main results.

Approximating the Time Optimal Solution for Accessible Unknown Systems by Learning using a Virtual Output

Time optimal control of systems with bounded inputs is a numerically awkward problem as essentially a whole trajectory has to be designed on the basis of a single final point at unknown time. Analytical solutions are possible only for very few problems, in general numerical techniques will be needed, which, in view of the non convexity of the associated optimization problem, will typically converge to a local minimum. In case of an unknown but experimentally accessible nonlinear system, such a solution cannot be found numerically, but a learning algorithm has been proposed which typically converges to a solution not far from the optimal one. However, as in the case of numerical computations, no guarantee of global optimality can be given. In the numerical case, different randomization techniques can be used to ascertain the existence of better solutions, e.g. by choosing different initial conditions. This is more difficult in the case of an experimental method for unknown systems, but this paper proposes an approach based on virtual outputs (linear combination of measurable states) for the same goal which is shown to work in a classical problem of counterintuitive time optimal control - the fastest climbing of a plane.

Regularity properties of time-optimal trajectories of an analytic single-input control-linear system in dimension three

We consider the problem of time-optimal control for systems of the form $$\dot x = f(x) + ug(x),x \in M,|u| \leqslant 1$$ , where the state spaceM is a three-dimensional real-analytic manifold,f andg are real-analytic vector fields onM, and admissible controls are scalar measurable functionsu(·) with values in ź1≤u≤1 a.e. We prove, for arbitraryf andg, that there exists an analytic subsetA ofM with positive codimension such that every point not inA has a neighborhoodU such that time-optimal trajectories that lie inU are, in nondegenerate cases, bang-bang with at most two switchings or concatenations of at most a bang arc, followed by a singular arc and another bang arc; in degenerate cases, wheneverq1 źU can be steered toq2źU in timeT inU, then there also exists a bang-bang trajectory with at most two switchings that steersq1 toq2 in timeT.

On the Local Structure of Time-Optimal Bang-Bang Trajectories in $\mathbb{R}^3$

We consider the problem of time-optimal control for systems of the form $\dot x = f(x) + ug(x)$ where f and g are smooth vector fields and admissible controls are measurable functions u with values in $ - 1 \leqq u \leqq 1$. Under the assumption that f, g and $[f,g]$ are independent, we prove that generically every point has a neighborhood $\mathcal{U}$ such that bang-bang trajectories that lie in $\mathcal{U}$ and have more than 7 switchings are not time-optimal.

Controllability and time-optimal control of systems with slow and fast modes

The controllability of linear systems with large and small time constants (singularly perturbed systems) is established. The time-optimal control of such systems is shown to be separable into two time scales related to the slow and fast modes of the system. An approximate design method of the time-optimal control is proposed, which is based on the separability of the fast and slow controls.