Sidewise Profile Control of 1-D Wave Equations with Variable Coefficients

1. Introduction

This paper addresses the sidewise boundary controllability problem for variable coefficients one-dimensional wave equations. The control acts on one extreme of the string with the objective that the solution tracks a given path or profile at the other free end. This sidewise profile control problem is also referred to as nodal profile or tracking control.

The problem is reformulated as a dual observability property for the corresponding adjoint system, which is proved using sidewise energy propagation arguments in a sufficiently large time, within the class of BV-coefficients. The research presents several open problems and perspectives for further investigation in this domain.

2. Problem Formulation

Consider the variable coefficients controlled 1-d wave equation:

ρ(x)y_tt - (a(x)y_x)_x = 0, 0 < x < L, 0 < t < T
y(x,0) = y⁰(x), y_t(x,0) = y¹(x), 0 < x < L
y(0,t) = u(t), y(L,t) = 0, 0 < t < T

Where T represents the time-horizon length, L is the string length, y = y(x,t) is the state, and u = u(t) is the control acting on the system through the extreme x = 0.

The coefficients ρ and a belong to BV and are uniformly bounded above and below by positive constants:

0 < ρ₀ ≤ ρ(x) ≤ ρ₁
0 < a₀ ≤ a(x) ≤ a₁ almost everywhere in (0,L)
ρ, a ∈ BV(0,L)

3. Mathematical Framework

The main objective is to analyze sidewise boundary controllability: Given a time-horizon T > 0, initial data y⁰(x), y¹(x), and a target profile p(t) for the flux at x = L, find u(t) such that the corresponding solution satisfies:

y_x(L,t) = p(t), t ≥ 0

This condition should hold in a time-subinterval of [0,T] under appropriate conditions on T, according to the velocity of wave propagation.

Due to finite velocity of propagation, this result doesn't hold for all T > 0 but only for T sufficiently large, allowing the control action at x = 0 to reach the other extreme x = L along characteristics.

4. Methodology

The approach involves reformulating the sidewise profile control problem as a dual observability property for the corresponding adjoint system. The proof employs sidewise energy propagation arguments within the class of BV-coefficients.

Key methodological elements include:

Dual Observability: Transforming the control problem into an observability problem for the adjoint system
Sidewise Energy Estimates: Utilizing energy propagation techniques to establish controllability
BV-Coefficient Analysis: Working within the bounded variation coefficient framework as the minimal regularity requirement
Characteristic Method: Accounting for finite velocity of wave propagation along characteristics

5. Main Results

The paper establishes several key results in sidewise profile controllability:

Regularity Requirements

BV represents the minimal regularity requirement for coefficients, with counterexamples existing in Hölder continuous classes

Time Constraints

Controllability requires sufficiently large time horizons to allow wave propagation from control to target boundary

Dual Framework

Successful reformulation of control problem as dual observability property for adjoint system

The research demonstrates that for coefficients slightly less regular than BV, weaker controllability properties emerge, requiring smoother initial data than expected in the BV framework.

6. Applications and Perspectives

Sidewise control problems have significant applications in various domains:

Gas Flow Networks: Motivated by applications in gas flow on networks, particularly nodal profile control problems
Quasilinear Hyperbolic Systems: Extension to 1-D quasilinear hyperbolic systems through constructive methods
Engineering Systems: Applications in mechanical systems, acoustic control, and structural dynamics

The paper identifies several open problems and research directions:

Extension to higher-dimensional wave equations
Analysis with less regular coefficients
Numerical implementation and computational aspects
Applications to more complex physical systems

Key Insights

Minimal Regularity

BV coefficients represent the minimal regularity requirement for achieving sidewise controllability in 1-D wave equations.

Finite Propagation

The finite velocity of wave propagation imposes natural constraints on the minimum time required for controllability.

Dual Approach

Reformulating control problems as dual observability problems provides powerful analytical tools for establishing controllability.

7. Conclusion

This research provides a comprehensive analysis of sidewise profile controllability for variable coefficient 1-D wave equations. The methodology based on dual observability and sidewise energy propagation arguments establishes controllability within the BV-coefficient framework under appropriate time constraints determined by wave propagation characteristics.

The results contribute significantly to the understanding of non-standard controllability problems where the objective is to track a given boundary profile rather than achieve a final state. The work opens several avenues for future research, particularly in extending these results to more complex systems and less regular coefficient classes.

The practical applications in gas flow networks and other physical systems highlight the relevance of these theoretical developments to real-world engineering problems.