[Paper Review] Control issues and linear projection constraints on the control and on the controlled trajectory

Sylvain Ervedoza|arXiv (Cornell University)|Sep 4, 2019

Stability and Controllability of Differential Equations22 references4 citations

TL;DR

This paper establishes conditions under which linear projection constraints can be imposed on both the control and the controlled trajectory in abstract linear control systems. It proves that approximate, exact, and null controllability with such constraints are achievable if a novel unique continuation property for the adjoint system holds, which is shown to be valid in various settings including parabolic and hyperbolic PDEs.

ABSTRACT

The goal of this article is to discuss controllability properties for an abstract linear system of the form $y' = Ay + Bu$ under some additional linear projection constraints on the control $u$ or / and on the controlled trajectory $y$. In particular, we discuss the possibility of imposing the linear projections of the controlled trajectory and of the control, in the context of approximate controllability, exact controllability and null-controllability. As it turns out, in all these settings, for being able to impose linear projection constraints on the trajectory and on the control, we will strongly rely on a unique continuation property for the adjoint system which, to our knowledge, has not been identified so far, and which does not seem classical. We shall therefore provide several instances in which this unique continuation property can be checked.

Motivation & Objective

• To investigate whether approximate, exact, and null controllability can be achieved when linear projection constraints are imposed on both the control input and the system trajectory.
• To identify the minimal structural conditions—specifically, a new unique continuation property for the adjoint system—under which such constrained controllability is possible.
• To extend classical controllability theory to settings where the control and trajectory must satisfy prescribed projections in function spaces.
• To provide verifiable conditions under which the novel unique continuation property holds, particularly in PDE contexts such as the heat and wave equations.
• To lay the foundation for practical applications involving insensitivity to parameter variations and bounded control sets, by analyzing the cost and feasibility of projection-constrained controls.

Proposed method

• Formulates the control problem for an abstract linear system $ y' = Ay + Bu $ in Hilbert spaces, with constraints on the control $ u $ and trajectory $ y $ via orthogonal projections onto closed subspaces $ \mathscr{G} \subset L^2(0,T;U) $ and $ \mathscr{W} \subset L^2(0,T;H) $.
• Introduces a new unique continuation property for the adjoint system: if the solution $ z $ of $ z' + A^*z = w $ satisfies $ B^*z = g $ in $ (0,T) $, then $ z_T = 0 $, $ g = 0 $, and $ w = 0 $, which is essential for constrained controllability.
• Applies duality arguments and functional analysis techniques to reduce the constrained controllability problems to the solvability of certain optimization problems involving the adjoint system.
• Uses variational methods to construct controls that satisfy the projection constraints exactly while achieving approximate, exact, or null controllability of the state.
• Validates the unique continuation property in concrete PDE models, including the heat equation with time-dependent potentials and the wave equation, using known Carleman estimates and spectral methods.
• Extends the framework to unbounded control operators and considers potential generalizations to Banach spaces and convex control constraints, suggesting future research directions.

Experimental results

Research questions

• RQ1 Under what conditions can a linear control system be approximately controllable while enforcing linear projection constraints on both the control and the trajectory?
• RQ2 What is the role of a novel unique continuation property for the adjoint system in enabling constrained controllability, and how can it be verified in practice?
• RQ3 Can exact or null controllability be achieved when the control and trajectory are required to project onto specific subspaces $ \mathscr{G} $ and $ \mathscr{W} $, respectively?
• RQ4 How does the cost of controlling the projections on $ \mathscr{G} $ and $ \mathscr{W} $ scale, and can this be quantified in terms of the system parameters?
• RQ5 To what extent can moment-based methods be adapted to solve constrained controllability problems when the control space is spanned by non-exponential functions?

Key findings

• Approximate controllability with projection constraints on both control and trajectory is possible if the novel unique continuation property for the adjoint system holds and $ \mathscr{W} $ is finite-dimensional.
• The key condition for constrained controllability is a unique continuation property for the adjoint system: if $ B^*z = g $ in $ (0,T) $ and $ z(T) = z_T $, then $ z_T = 0 $, $ g = 0 $, and $ w = 0 $, which ensures the existence of a control satisfying the constraints.
• The unique continuation property is verified for the heat equation with time-dependent potentials and for the wave equation under suitable geometric and spectral conditions.
• The results extend to unbounded control operators when $ B $ is admissible, preserving the validity of the main theorems.
• The framework allows for constraints on the control to lie in arbitrary closed subspaces $ \mathscr{G} $, including finite-dimensional spaces such as span of exponential functions.
• The paper identifies open problems related to moment-based control methods and cost quantification, particularly for non-exponential control subspaces, suggesting promising directions for future research.