[Paper Review] Frequency theorem for the regulator problem with unbounded cost functional and its applications to nonlinear delay equations

Mikhail Anikushin|arXiv (Cornell University)|Mar 27, 2020

Stability and Controllability of Differential Equations32 references9 citations

TL;DR

This paper establishes a non-singular frequency theorem for the regulator problem with unbounded cost functionals in a rigged Hilbert space setting, enabling the derivation of a Lyapunov-like functional via a bounded linear operator. It applies this framework to derive frequency-domain stability criteria for non-autonomous nonlinear delay equations, recovering the circle criterion in special cases.

ABSTRACT

We study the regulator problem with an unbounded cost functional of a general type. A motivation comes from delay equations, which has the feedback part with discrete delays (or, in other words, delta-like measurements, which are unbounded in $L_{2}$). We treat the problem in an abstract context of a certain Hilbert space, which is rigged by a Banach space. We obtain a version of the non-singular frequency theorem, which guarantees the existence of a unique optimal process, starting in the Banach space. We show that the optimal cost (that is the value of the functional on the optimal process) is given by the quadratic of a bounded linear operator from the Banach space to its dual and this form can be used as a Lyapunov-like functional. For a large class of non-autonomous nonlinear delay equations in a feedback form we obtain a frequency-domain stability criteria, which in particular cases coincides with the well-known circle criterion.

Motivation & Objective

• To address the regulator problem with an unbounded cost functional arising in systems with discrete delays, such as those with delta-like measurements.
• To establish the existence and uniqueness of an optimal process starting in a Banach space within a rigged Hilbert space framework.
• To characterize the optimal cost as a quadratic form of a bounded linear operator from the Banach space to its dual, enabling Lyapunov-like stability analysis.
• To derive frequency-domain stability criteria for a broad class of non-autonomous nonlinear delay equations in feedback form.
• To show that the derived criteria generalize and include the classical circle criterion as a special case.

Proposed method

• Formulates the regulator problem in a rigged Hilbert space, where the state space is a Hilbert space and the initial states lie in a Banach space continuously embedded in it.
• Introduces a general unbounded cost functional that captures the energy of feedback with discrete delays, modeling delta-like measurements.
• Applies an abstract frequency theorem to guarantee the existence and uniqueness of the optimal process in the Banach space setting.
• Derives the optimal cost as a quadratic form involving a bounded linear operator from the Banach space to its dual, which serves as a Lyapunov-like functional.
• Translates the stability analysis into the frequency domain by analyzing the spectral properties of the system operator and its resolvent.
• Applies the frequency-domain criteria to nonlinear delay equations with feedback, yielding conditions for asymptotic stability.

Experimental results

Research questions

• RQ1 Under what conditions does the regulator problem with an unbounded cost functional admit a unique optimal process when the initial state lies in a Banach space?
• RQ2 How can the optimal cost be represented in a way that enables Lyapunov-type stability analysis for delay systems?
• RQ3 What frequency-domain conditions ensure the asymptotic stability of non-autonomous nonlinear delay equations with feedback?
• RQ4 In what sense does the derived stability criterion generalize or reduce to the classical circle criterion?
• RQ5 Can the abstract framework be applied to systems with discrete delays modeled by unbounded operators in L2?

Key findings

• The optimal process exists and is unique for initial states in the Banach space, under the proposed abstract framework.
• The optimal cost is represented as a quadratic form of a bounded linear operator from the Banach space to its dual, providing a Lyapunov-like functional.
• The stability of non-autonomous nonlinear delay equations in feedback form is characterized by a frequency-domain criterion derived from the frequency theorem.
• The derived stability criterion reduces to the classical circle criterion in specific cases, such as linear time-invariant systems with scalar delays.
• The framework accommodates unbounded cost functionals arising from discrete delays, which are unbounded in L2, by embedding the state space in a rigged Hilbert space.
• The method enables stability analysis without requiring the system to be autonomous or linear, extending applicability to broader classes of delay equations.