The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

doi:10.1016/j.sysconle.2008.05.006

Systems & Control Letters

Volume 57, Issue 11, November 2008, Pages 964-970

https://doi.org/10.1016/j.sysconle.2008.05.006 Get rights and content

Abstract

We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle.

Introduction

In a broad sense, hybrid control systems are control systems involving both continuous and discrete variables. In recent years, optimization problems for hybrid systems have attracted a significant attention of specialists in control. One of the most important results in the study of such problems is the Hybrid Maximum Principle proved in [4], [5]. This proof is rather difficult, since it follows the standard line of the full procedure of direct proof of Maximum Principle (MP), based on the introduction of special class of control variations (e.g., needle-like ones), calculation of the increments of the cost and all constraints, etc. As is well known, this procedure is very laboursome even in the case of the classical optimal control problem without discrete variables. However, as will be shown below, one does not need to perform all this complicated procedure for obtaining the hybrid MP if one supposes that the Pontryagin MP is known for the standard optimal control problem with nonseparated terminal constraints. After some transformation of the hybrid problem, MP for it is an easy consequence of the classical Pontryagin MP.

The statement of the hybrid optimal control problem assumes the presence of a finite number of control systems, each of which is defined on its own space of variables (possibly of different dimensions). A trajectory moving under one of these systems, at some moment of time can switch to any other system, and it can do so a finite number of times. The hybridity in such systems just signifies the presence of both continuous and discontinuous dynamics of state variables. One needs to choose a sequence of control systems, durations of motion under each system, and a control variable for each system that minimize the given cost functional.

The sequence of control systems under which a trajectory moves is not defined a priori. However, when investigating a given trajectory for optimality, we assume this sequence defined and fixed, like in other papers we are aware of. (The thing is that variations of this sequence generate trajectories which are “far” from the given one, and hence they cannot be compared by methods of analysis.)

Section snippets

Statement of the problem

Let $t_{0} < t_{1} < \dots < t_{ν}$ be real numbers. Denote by $Δ_{k}$ the time interval $[t_{k - 1}, t_{k}]$ . For any collection of continuous functions $x_{k} : Δ_{k} \to R^{n_{k}}$ , $k = 1, \dots, ν$ , define a vector $p = (t_{0}, (t_{1}, x_{1} (t_{0}), x_{1} (t_{1})), (t_{2}, x_{2} (t_{1}), x_{2} (t_{2})), \dots, (t_{ν}, x_{ν} (t_{ν - 1}), x_{ν} (t_{ν})))$ of dimension $d = 1 + ν + 2 \sum_{k = 1}^{ν} n_{k}$ .

On the time interval $Δ = [t_{0}, t_{ν}]$ consider the optimal control problem $Problem A : {\begin{cases} {\dot{x}}_{k} = f_{k} (t, x_{k}, u_{k}), u_{k} \in U_{k}, for t \in Δ_{k}, k = 1, \dots, ν, \\ η_{j} (p) = 0, j = 1, \dots, q, \\ φ_{i} (p) \leq 0, i = 1, \dots, m, \\ J = φ_{0} (p) \to min, \end{cases}$ where $x_{k} \in R^{n_{k}}, u_{k} \in R^{r_{k}}$ , the functions $x_{k} (t)$ are absolutely continuous, $u_{k} (t)$ are

Obtaining the hybrid maximum principle

We will pass from Problem A to some problem of canonical type $K$ and establish a correspondence between the admissible and optimal processes in these problems. The idea of such passage is quite natural: one should reduce all the state and control variables to a common fixed time interval, for example, to [0, 1].

Let $(θ, x (t), u (t))$ be an arbitrary admissible process in Problem A.

Introduce a new time $τ \in [0, 1]$ and define functions $ρ_{k} : [0, 1] \to Δ_{k}$ , $k = 1, \dots, ν$ , from the equations: $\frac{d ρ_{k}}{d τ} = z_{k} (τ), ρ_{k} (0) = t_{k - 1},$ where

Hybrid systems with a quasivariable control set

Along with the hybrid control problems of the above “standard” type A, in literature there are also considered hybrid systems in which the control set $U_{k}$ on each time interval $Δ_{k}$ depends, in a special way, on the values of state variable at switching times $t_{k}$ , namely, the control set on $Δ_{k}$ is $σ_{k} (p) U_{k}$ , where the functions $σ_{k} (p)$ have the derivative on $P$ . We will refer to these problems as problems with quasivariable control set or, for the sake of brevity, problems of type $B$ . Such a problem was

Example: Control of a car with two gears [6]

A car moves under the law $\dot{x} = y, \dot{y} = u g_{1} (y)$ , $u \in U$ on the time interval $Δ_{1} = [0, t_{1}]$ , and under the law $\dot{x} = y, \dot{y} = u g_{2} (y), u \in σ (y (t_{1})) U$ on the time interval $Δ_{2} = [t_{1}, T]$ .

The initial and final time moments $t_{0} = 0$ and $t_{2} = T$ are fixed, the moment $t_{1}$ is not fixed, the set $U = [0, 1]$ , the functions $g_{1}, g_{2}, σ$ are positive and differentiable in $R^{1}$ . The car starts from the point $(x^{0}, y^{0}) = (0, 0)$ , and the state variables $x$ and $y$ are assumed to be continuous on the whole interval $Δ = [0, T]$ . It is required to maximize $x (T)$ .

Rewrite

Acknowledgments

This work was supported by the Russian Foundation for Basic Research, project no. 08-01-00685. The authors thank Boris Miller and Vladimir Dykhta for valuable discussions, and Hector Sussmann for his interest and attention to this work.

References (8)

L.S. Pontryagin et al.
Mathematical Theory of Optimal Processes
(1961)
V.M. Alekseev et al.
Optimal Control
(1979)
U.M. Volin et al.
Maximum principle for discontinuous systems and its application to problems with state constraints
Izvestia vuzov. Radiophysics
(1969)
H.J. Sussmann, A maximum principle for hybrid optimal control problems, in: Proc. of 38th IEEE Conference on Decision...

There are more references available in the full text version of this article.

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The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

Abstract

Introduction

Section snippets

Statement of the problem

Obtaining the hybrid maximum principle

Hybrid systems with a quasivariable control set

Example: Control of a car with two gears [6]

Acknowledgments

Mathematical Theory of Optimal Processes

Optimal Control

Maximum principle for discontinuous systems and its application to problems with state constraints

Izvestia vuzov. Radiophysics