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A Method of the Quasidifferential Descent in a Problem of Bringing a Nonsmooth System from One Point to Another

Alexander Fominyh

TL;DR

This work tackles the challenge of steering a system with a quasidifferentiable right-hand side from an initial to a final state within finite time using piecewise-continuous controls. It introduces a novel separation of the phase trajectory $x(t)$ from its derivative $z(t)$, enforced by a specially crafted penalty, transforming the problem into an unconstrained variational problem in the space $X=C_n[0,T]\times P_n[0,T]\times P_m[0,T]$. The core contribution is a pointwise quasidifferential framework that yields descent directions via a per-time Hausdorff-distance subproblem, enabling a quasidifferential descent method (MQD) whose directions can be computed independently at discretization points. Numerical examples demonstrate the method's capability to handle nonsmooth dynamics and constraints, achieving accurate endpoint conditions with reasonable computational effort and suggesting parallelizable, direct optimization advantages over traditional discretization approaches.

Abstract

The paper considers the problem of constructing program control for an object described by a system with a quasidifferentiable right-hand side. The control aim is to bring the system from a given initial position to a given final state in given finite time. The admissible controls are piecewise continuous vector-functions with values from a parallelepiped. The original problem is reduced to unconditional minimization of a functional. Herewith, the new technical idea is implemented to consider phase trajectory and its derivative as independent variables (and to take the natural relation between them into account via a special penalty function). This idea qualitatively simplified the quasidifferential structure and allowed to overcome the principal difficulties in constructing the steepest descent direction. The quasidifferentiability of the functional is proved, necessary conditions for its minimum are obtained in terms of quasidifferential. In contrast to the existing ones, due to the mentioned idea to ``separate'' the trajectory and its derivative the obtained optimality conditions in the paper are pointwise. In order to solve the obtained minimization problem in the functional space the quasidifferential descent method is applied. Then the discretization is implemented. In contrast to majority of existing methods when the initial problem is discretized, here the discretization is implemented after the quasidifferential is already obtained. The quasidifferential descent directions are calculated independently at each time moment of discretization due to the comparatively simple quasidifferential structure, possible to obtain via the technical idea noted. The algorithm developed is demonstrated by examples. The proposed method can be applied to nonsmooth optimal control problem in Lagrange form (additionally the integral with a quasidifferentiable integrand is to be minimized).

A Method of the Quasidifferential Descent in a Problem of Bringing a Nonsmooth System from One Point to Another

TL;DR

This work tackles the challenge of steering a system with a quasidifferentiable right-hand side from an initial to a final state within finite time using piecewise-continuous controls. It introduces a novel separation of the phase trajectory from its derivative , enforced by a specially crafted penalty, transforming the problem into an unconstrained variational problem in the space . The core contribution is a pointwise quasidifferential framework that yields descent directions via a per-time Hausdorff-distance subproblem, enabling a quasidifferential descent method (MQD) whose directions can be computed independently at discretization points. Numerical examples demonstrate the method's capability to handle nonsmooth dynamics and constraints, achieving accurate endpoint conditions with reasonable computational effort and suggesting parallelizable, direct optimization advantages over traditional discretization approaches.

Abstract

The paper considers the problem of constructing program control for an object described by a system with a quasidifferentiable right-hand side. The control aim is to bring the system from a given initial position to a given final state in given finite time. The admissible controls are piecewise continuous vector-functions with values from a parallelepiped. The original problem is reduced to unconditional minimization of a functional. Herewith, the new technical idea is implemented to consider phase trajectory and its derivative as independent variables (and to take the natural relation between them into account via a special penalty function). This idea qualitatively simplified the quasidifferential structure and allowed to overcome the principal difficulties in constructing the steepest descent direction. The quasidifferentiability of the functional is proved, necessary conditions for its minimum are obtained in terms of quasidifferential. In contrast to the existing ones, due to the mentioned idea to ``separate'' the trajectory and its derivative the obtained optimality conditions in the paper are pointwise. In order to solve the obtained minimization problem in the functional space the quasidifferential descent method is applied. Then the discretization is implemented. In contrast to majority of existing methods when the initial problem is discretized, here the discretization is implemented after the quasidifferential is already obtained. The quasidifferential descent directions are calculated independently at each time moment of discretization due to the comparatively simple quasidifferential structure, possible to obtain via the technical idea noted. The algorithm developed is demonstrated by examples. The proposed method can be applied to nonsmooth optimal control problem in Lagrange form (additionally the integral with a quasidifferentiable integrand is to be minimized).
Paper Structure (9 sections, 4 theorems, 126 equations, 7 figures)

This paper contains 9 sections, 4 theorems, 126 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions and Notation
  3. Statement of the Problem
  4. Reduction to a Variational Problem
  5. Necessary Minimum Conditions
  6. Quasidifferential Descent Method
  7. Numerical Examples
  8. Discussion
  9. Conclusion

Key Result

Theorem 5.1

Let Assumptions 5.2.1, 5.2.2 be satisfied. In order for the point $(x^*, z^*, u^*)$ to minimize the functional $I(x, z, u)$, it is necessary and sufficient to have $I(x^*, z^*, u^*) =~0$.

Figures (7)

  • Figure 1: Example \ref{['ex_4.2']}, control and trajectories on iterations: 2, 9, 13, 17
  • Figure 2: Example \ref{['ex_4.3']}, control on iterations: 2, 13, 29, 45, 58
  • Figure 3: Example \ref{['ex_4.3']}, trajectories on iterations: 2, 13, 29, 45, 58
  • Figure 4: Example \ref{['ex_4.4']}, the resulting control and trajectories
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (19)

  • Remark 1
  • Theorem 5.1
  • Theorem 5.2
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 6.1
  • Lemma 6.2
  • Remark 5
  • Remark 6
  • ...and 9 more