[Paper Review] Optimal Switching for Hybrid Semilinear Evolutions

Fabian Rueffler, Falk M. Hante|arXiv (Cornell University)|May 17, 2016

Advanced Thermodynamics and Statistical Mechanics1 references3 citations

TL;DR

This paper develops necessary optimality conditions for hybrid semilinear evolution systems that switch between different modes governed by nonlinearly perturbed strongly continuous semigroups, including state resets and switching costs. It derives gradient representations via adjoint equations for switching times and mode-insertions, generalizing ODE-based methods to PDEs and delay equations with switching in the principal part or generator.

ABSTRACT

We consider the optimization of a dynamical system by switching at discrete time points between abstract evolution equations composed by nonlinearly perturbed strongly continuous semigroups, nonlinear state reset maps at mode transition times and Lagrange-type cost functions including switching costs. In particular, for a fixed sequence of modes, we derive necessary optimality conditions using an adjoint equation based representation for the gradient of the costs with respect to the switching times. For optimization with respect to the mode sequence, we discuss a mode-insertion gradient. The theory unifies and generalizes similar approaches for evolutions governed by ordinary and delay differential equations. More importantly, it also applies to systems governed by semilinear partial differential equations including switching the principle part. Examples from each of these system classes are discussed.

Motivation & Objective

• To develop a unified framework for optimal switching in hybrid systems governed by abstract semilinear evolutions.
• To address the lack of gradient-based optimization methods for systems with switching in the generator, such as PDEs with changing principal parts.
• To extend switching-time optimization and mode-insertion gradients from ODEs to infinite-dimensional systems with nonlinear perturbations and state resets.
• To provide differentiable cost representations using adjoint equations for both switching times and mode sequences.
• To enable gradient-based optimization in complex systems like gas networks, where switching affects the evolution operator.

Proposed method

• Derives the gradient of the cost functional with respect to switching times using an adjoint equation formulation in a Banach space setting.
• Introduces a mode-insertion gradient to optimize over the sequence of modes, enabling algorithmic exploration of different mode orders.
• Uses semigroup theory to model continuous dynamics under different modes, with generators $A^j$ generating $C^0$-semigroups.
• Incorporates nonlinear state reset maps $g^{j,j'}$ at switching times and Lagrange-type cost functions with switching penalties.
• Applies the theory to systems with switching in the principal part, such as transport vs. diffusion PDEs or delay parameters in DDEs.
• Establishes differentiability of the cost functional under mild regularity assumptions on the system parameters and semigroup generators.

Experimental results

Research questions

• RQ1 How can the gradient of the cost functional be represented with respect to switching times in hybrid semilinear evolution systems with state resets and switching costs?
• RQ2 What is the appropriate gradient formulation for optimizing over the sequence of modes, particularly for inserting a new mode into an existing sequence?
• RQ3 Can the adjoint-based gradient approach for ODEs be generalized to systems governed by PDEs where the generator itself switches between modes?
• RQ4 How does the theory handle switching in the principal part of a PDE, such as switching between transport and diffusion operators?
• RQ5 What conditions ensure the differentiability of the cost functional and the validity of the adjoint representation in infinite-dimensional settings?

Key findings

• The gradient of the cost with respect to switching times is represented via the solution of an adjoint equation, enabling gradient-based optimization.
• The mode-insertion gradient allows for systematic exploration of different mode sequences, forming the basis for hybrid optimization algorithms.
• The theory generalizes ODE-based switching-time optimization to PDEs and delay equations, including cases where the generator (e.g., the principal part) switches.
• For a system switching from a transport equation ($A^1 = \partial_x$) to a diffusion equation ($A^2 = \partial_x^2$), the optimal switching time is $\tau = 0$, as shown by a non-negative gradient.
• The adjoint solution propagates backward in time and enables efficient computation of sensitivity, even when the semigroups do not commute.
• The framework is consistent with stability analysis and can be extended to unbounded perturbations and general boundary conditions in future work.