Non-Causal Boundary Predictor in PDE Control

• The paper introduces a finite-dimensional predictor that compensates for constant input delays via Artstein reduction and pole-shifting techniques.
• It employs spectral modal decomposition to isolate unstable modes, ensuring exponential stabilization in the H¹-norm.
• The approach streamlines real-time implementation by using only current and historical control data, contrasting with backstepping methods.

A non-causal boundary predictor is a feedback control structure for boundary-delayed parabolic partial differential equations (PDEs), specifically tailored to stabilize these systems in the presence of constant input delays. Such predictors are termed “non-causal” because a naïve implementation would seemingly require future state knowledge; however, through integral memory transformation (specifically, via the Artstein reduction), a practical, causal form is achieved that utilizes only present and historical data. The finite-dimensional predictor-based approach is designed by isolating unstable modes of a modal decomposition and applying pole-shifting techniques from classical control. This framework is rigorously developed for the 1D reaction-diffusion equation with delayed boundary actuation, where the predictor serves as an efficient, implementable compensator that guarantees exponential stabilization in the $H^1$-norm (Bresch-Pietri et al., 2015).

1. Boundary-Delayed Reaction-Diffusion Equations

The problem setting is the control of a 1D reaction-diffusion PDE on the domain $(0,L)$, with Dirichlet boundary actuation at $x=L$ that is subject to a constant known input delay $D>0$:

$$\begin{cases} y_t(t,x) = y_{xx}(t,x) + c(x) y(t,x), & x\in(0,L),\; t>0, \ y(t,0) = 0,\quad y(t,L) = u_p(t) = u(t-D), & t>0, \ y(0,x) = y_0(x),\;\; u_p(t) = 0 \;\forall\, t\in[0,D]. \end{cases}$$

Here, $c\in L^\infty(0,L)$ denotes the reaction potential. The delayed control $u_p(t)$ is the main source of instability since direct feedback using past controls introduces non-minimum phase behavior.

Homogenization at the boundary is achieved by defining $w(t,x) := y(t,x) - u_p(t)$, yielding homogeneous Dirichlet conditions and a PDE-ODE cascade structure involving both the system state and the delayed boundary control.

2. Spectral Analysis and Finite-Dimensional Truncation

Eigenfunction decomposition of the principal operator $A = \frac{d^2}{dx^2} + c(x)$, with homogeneous Dirichlet boundary conditions, provides an orthonormal basis $\{e_j(x)\}_{j\geq 1}$ with associated eigenvalues $\lambda_1 \geq \lambda_2 \geq \dots \to -\infty$.

Expanding the state as $w(t,x) = \sum_{j=1}^{\infty} w_j(t)e_j(x)$, the dynamics reduce to a countable ODE system with delay embedded in the input channel:

$$\begin{cases} u_p'(t) = \alpha(t), \ w_j'(t) = \lambda_j w_j(t) + a_j u_p(t) + b_j \alpha(t), \quad j=1,2,\dots, \end{cases}$$

where $a_j = \int_0^L c(x) e_j(x) dx$, $b_j = -\int_0^L e_j(x) dx$, and $\alpha(t) := u_p'(t)$.

A critical observation is that only finitely many modes have non-negative eigenvalues (potentially unstable), so the system can be decomposed into a finite-dimensional “unstable” part

$$X_1(t) := \begin{pmatrix} u_p(t) & w_1(t) & \dots & w_n(t) \end{pmatrix}^T,$$

with matrix dynamics

$X_1'(t) = A_1 X_1(t) + B_1 \alpha(t-D).$

3. Artstein Reduction and Predictor Feedback

To treat the delay, the Artstein transformation is employed. The new state

$$Z_1(t) = X_1(t) + \int_{t-D}^t e^{(t-s-D)A_1} B_1 \alpha(s) ds$$

obeys a delay-free ODE: $Z_1'(t) = A_1 Z_1(t) + e^{-DA_1} B_1 \alpha(t).$ This recasts the control design into a conventional linear, finite-dimensional framework.

Choosing a static gain vector $K_1$ so that $A_1 + e^{-DA_1}B_1K_1$ is Hurwitz (using the Kalman–pole-shifting theorem, e.g., placing all eigenvalues at $-1$), the delay-free law

$\alpha(t) = K_1 Z_1(t)$

ensures exponential convergence of $Z_1(t)$.

The practical, causal feedback—the non-causal boundary predictor—emerges by expressing $\alpha(t)$ in terms of current and past states: $$U(t) = K_1 \left( X_1(t) + \int_{t-D}^t e^{(t-s-D)A_1} B_1 U(s) ds \right), \quad t \geq D,$$ with $U(t) = 0$ for $t \in [0,D)$, where $U(\cdot)$ is the implemented control signal.

4. Gain Design and Pole Placement

The structure $(A_1, e^{-DA_1}B_1)$ is Kalman-controllable. As such, closed-loop pole assignment is feasible by selecting

$\det(sI - (A_1 + e^{-DA_1}B_1K_1)) = (s+1)^{n+1},$

so all unstable eigenvalues are moved to the desired negative location. The gain vector $K_1(D)$ is calculated, for instance, via companion-form analysis, yielding a unique stabilizer in $\mathbb{R}^{1 \times (n+1)}$.

5. Exponential Stability and Lyapunov Analysis

A composite Lyapunov functional guarantees exponential stability in the full system, including unmodeled infinite-dimensional stable modes. The functional

$$V_D(t) = M \{ Z_1(t)^T P Z_1(t) + \int_{t-D}^t Z_1(s)^T P Z_1(s) ds \} - \tfrac{1}{2} \langle w(t), Aw(t) \rangle_{L^2(0,L)}$$

incorporates a positive definite solution $P$ to the Lyapunov equation

$$P(A_1+e^{-DA_1}B_1K_1) + (A_1+e^{-DA_1}B_1K_1)^T P = -I,$$

and $M > 0$ is chosen suitably large to dominate the stable residual. Exponential decay

$$\dot V_D(t) \leq -\mu V_D(t) \implies V_D(t) \leq V_D(0) e^{-\mu t}$$

ensures that $\|y(t,\cdot)\|_{H^1(0,L)}$ converges to zero uniformly with respect to the delay parameter $D$.

6. Comparison with Backstepping and Significance

The predictor-based law circumvents direct non-causality (dependence on future state $X_1(t+D)$) by encoding the prediction task as a Volterra integral of past controls, which is always available in real time. In contrast, the backstepping approach entails full-state Volterra transformations and kernel PDEs for boundary delay compensation, which may complicate implementation.

Key distinctions:

Aspect	Predictor-based (finite-dim)	Backstepping (PDE)
Modal focus	Unstable modes only	Full-state transformation
Implementation	Linear ODE pole-placement, Volterra int.	Solution/inversion of kernel PDE
Real-time data needed	Present state, control history	Observer for distributed kernel

The non-causal boundary predictor’s implementation is considerably streamlined, relying on spectral truncation and conventional linear control methods, and does not require full inversion of operator transforms. This suggests significant applicability to higher-dimensional and more complex settings where backstepping may be computationally demanding.

7. Context, Extensions, and Implications

The finite-dimensional predictor-based feedback framework, as rigorously developed by Bresch-Pietri, Prieur, and Trélat, serves as a paradigm for delay compensation and stabilization in boundary-controlled parabolic systems. The Volterra-memory realization of delay compensation generalizes to broader classes of boundary input delays and may influence designs for interconnected PDE-ODE cascades and distributed parameter systems (Bresch-Pietri et al., 2015). A plausible implication is the strategy’s adaptability to multi-dimensional reaction-diffusion systems and modular hybrid control architectures, where precise modal truncation and pole placement can be obtained. The approach establishes theoretical equivalence, in closed-loop stabilization, to more traditional infinite-dimensional backstepping schemes while offering implementation advantages.