Boundary Control and Observation Systems

• Boundary control and observation systems are operator-theoretic frameworks that manage infinite-dimensional dynamics using boundary actuation and sensing.
• They ensure well-posedness and stability through energy estimates, semigroup theory, and stringent admissibility conditions.
• Applications span beam and string models, networked flows, and industrial systems, employing robust and adaptive control strategies.

Boundary control and observation systems refer to operator-theoretic frameworks for modeling, analysis, and synthesis of feedback interconnections for partial differential equations (PDEs) and infinite-dimensional dynamical systems where actuation and sensing are implemented via the boundary of the spatial domain. These systems encompass a rigorous treatment of physical systems—such as vibrating strings, elastic beams, diffusive or transport processes, and wave equations—where only the "boundary" is accessible for control and measurement. The topic involves intricate challenges due to the unboundedness of the control/observation operators, well-posedness subtleties, passivity and dissipativity structures, as well as finite- and infinite-dimensional control design methodologies.

1. Mathematical Formulation and Operator Framework

Boundary control and observation systems (BCOS) are generally operated over spaces such as $X$, $U$, and $Y$, where $X$ is the state space (typically $L^2$-type for energy), $U$ is the control input space, and $Y$ is the observation (output) space. The core formulation consists of:

• State Equation: $\dot{x}(t) = A x(t)$, where $A: D(A) \subset X \to X$ is an unbounded (often differential) operator.
• Boundary Control Operator: $B : D(B) \subset X \to U$, modeling actuation at some trace or boundary of the domain.
• Boundary Observation Operator: $C : D(C) \subset X \to Y$, providing measurement at the physical boundary.

Classical BCOS structure is

$$\dot{x}(t) = A x(t),\quad B x(t) = u(t),\quad y(t) = C x(t),\quad x(0) = x_0.$$

The relevant solution concepts are

• Classical Solutions: $x(t) \in C^1([0,T]; X)$, $x(t) \in D(A)$.
• Mild Solutions: $x(t) = T(t)x_0 + \int_0^t T_{-1}(t-s) B u(s) ds$ when $T$ is a $C_0$-semigroup and $T_{-1}$ is the extrapolated semigroup on $X_{-1}$ (Elghazi et al., 28 Jan 2025).

Key conditions:

• $A$ with $D(A) \cap \ker B$ generates a $C_0$-semigroup on $X$.
• There exists a bounded right-inverse $B_0$ of $B$.
• $C$ is graph-norm bounded on $D(A)$ (Elghazi et al., 4 Jan 2026, Elghazi et al., 28 Jan 2025).

Frameworks such as the abstract Dirichlet map, admissibility in the sense of Weiss–Staffans, and energy estimates via passivity underlie the analysis (Engel et al., 2016, Jacob et al., 2019).

2. Well-Posedness and Stability

The concept of well-posedness in BCOS requires existence, uniqueness, and continuous dependence of solutions, typically formulated in terms of an input–output estimate: $$\|x(t)\|^2_X + \int_0^t \|y(s)\|^2_Y ds \leq m \left( \|x_0\|_X^2 + \int_0^t \|u(s)\|^2_U ds \right),$$ for some $t, m > 0$ and all classical solutions (Elghazi et al., 4 Jan 2026).

A sharp, verifiable condition for well-posedness is often characterized by invertibility of a boundary matrix $B_1$ associated with the highest-derivative boundary variables in port-Hamiltonian systems (Elghazi et al., 28 Jan 2025). In one-dimensional, second-order systems, $B_1$ acts on the "flow" part of the boundary port variables, and invertibility is necessary and sufficient for the existence of an admissible feedback and thus for well-posedness.

Scattering-passivity inequalities,

$$2 \Re \langle A x, x \rangle_X \leq \langle R u, u \rangle_U - \langle J C x, C x \rangle_Y,$$

ensure dissipativity, energy decay, and bound the input–state–output operator. These dissipative structures tie directly to well-posedness via energy methods and Lyapunov functionals (Jacob et al., 2019).

Well-posedness is a prerequisite for establishing the existence of transfer functions, well-defined regulator equations, and robust or optimal control in infinite-dimensional settings (Elghazi et al., 4 Jan 2026, Elghazi et al., 28 Jan 2025).

3. Controllability and Observability

In BCOS, controllability and observability are formulated in terms of the reachability map and dual observability map, inherently linked to the semigroup generated by $A$ and the admissibility of $B$ and $C$:

• Exact Controllability: The reachable subspace at time $T$,

$$\mathcal{R}(T) = \operatorname{Ran} \mathcal{B}_T, \quad \mathcal{B}_T u = ( \lambda - A ) \int_0^T T(T-s) B u(s) ds,$$

spans $X$ for some $T > 0$ (Engel et al., 2016).

• Exact Observability: There exists $T > 0$ such that

$$\int_0^T \| C T(t) x_0 \|_Y^2 dt \geq m \| x_0 \|_X^2$$

for some $m > 0$ and all $x_0 \in X$ (Elghazi et al., 4 Jan 2026).

• Operator-Hautus tests and the invariance of reachability/observability under admissible boundary static feedbacks apply (Elghazi et al., 4 Jan 2026).

The analysis extends to positive controllability and approximate reachability via lattice and cone-theoretic methods, especially relevant for transport on networks and systems with sign constraints (Engel et al., 2016).

4. System Classes, Structure-Preserving Discretization, and Interconnection

A prominent subclass is port-Hamiltonian systems (PHS), particularly in one dimension, whose structure naturally encodes energy, passivity, and interconnection properties. The state equation takes the form: $$\dot{z}(ζ,t) = P_1 ∂_ζ (L(ζ) z(ζ,t)) + P_0 L(ζ) z(ζ,t),$$ with $P_1$ skew-symmetric, $L(ζ)$ positive-definite, and boundary port variables defined as linear combinations of $H z$ and $P_1 H z$ at the domain endpoints (Toledo et al., 2019).

Structure-preserving discretization—via mixed finite differences or finite elements on staggered grids—ensures the discrete port-Hamiltonian structure, preservation of passivity, and convergence of energy norms, spectra, and transfer functions. This approach underpins "early lumping" designs for observer-based boundary control, where finite-dimensional controllers (e.g., LQR, pole-placement) are synthesized on the truncated model and then interconnected with the original infinite-dimensional plant via matching conditions to guarantee closed-loop stability (Toledo et al., 2019).

In contrast, "late lumping" designs maintain infinite-dimensional feedback laws from exact backstepping or flatness-based techniques, subsequently projecting and truncating only the bounded component of the gains. The spectral convergence theorem ensures that, under weak conditions (e.g., Riesz-spectral target generator), the finite-dimensional realizations maintain the desired closed-loop spectrum and exponential stability (Riesmeier et al., 2022).

5. Control, Estimation, and Robustness Methodologies

Boundary control and observation admit multiple control design and estimation methodologies, with an emphasis on feedback that acts directly at the boundary:

• State and Output Feedback: Observer-based output-feedback controllers may be constructed via decoupled Luenberger observers in the finite-dimensional setting (Toledo et al., 2019).
• Backstepping and Flatness-Based Designs: Infinite-dimensional transformations (Volterra or shift-differentiation) convert the system to a target form with desired stability, with exact or approximate realization depending on actuator/observer structure (Riesmeier et al., 2022, Alalabi et al., 2023).
• Robust Output Regulation and the Internal Model Principle: For infinite-dimensional output spaces, exact robust regulation requires infinite-dimensional controllers, motivating finite-dimensional approximations for practical implementation; such designs rely on G-conditions for the internal model (Humaloja et al., 2017). Approximate robust output regulation is achieved by projecting onto finite-dimensional subspaces and designing controllers that ensure the projected output converges within prescribed tolerances under small perturbations (Humaloja et al., 2017, Phan et al., 2019).
• Nonlinear Adaptive Control: Funnel control for BCOS achieves prescribed transient performance and tracking by an adaptive, output-only boundary law ensuring $\| e(t) \| < 1/\varphi(t)$, with global well-posedness resulting from dissipativity and m-dissipative operator theory (Puche et al., 2019).
• Restricted Observability and Partial Feedback: Feedback stabilization remains achievable under partial boundary measurement (restricted observability), provided the gain matrix satisfies dissipativity inequalities in the observed sub-blocks, but may preclude stabilization otherwise (Banda et al., 27 Jan 2025).
• Robustness Analysis: The $\nu$-metric provides a quantitative measure of robust stability for PDEs with parametric boundary uncertainty, connecting analytic factorization in Hardy spaces with frequency-response chordal distances (Sasane, 12 Dec 2025).

6. Advanced Topics and Extensions

• Stochastic BCOS: Stochastic boundary control and observation systems are established via admissibility theory for unbounded (boundary) operators, ensuring unique mild solutions and input–output well-posedness for SPDEs (e.g., stochastic heat, Schrödinger equations) (Lu, 2015, Hadd et al., 2021). The framework also accommodates stochastic boundary input delays through semigroup-product reformulations and Yosida extensions, guaranteeing state/output $L^2$-norm estimates and supporting delay-robust observer/controller design (Hadd et al., 2021).
• Fractional and Nonlocal Systems: BCOS theory extends to fractional parabolic and wave PDEs, where critical observability and controllability properties depend on spectral decay and the fractional exponent $s > 1/2$, analyzed via Pohozaev-type identities and Lebeau–Robbiano iteration (Biccari et al., 24 Apr 2025).
• Abstract Boundary Data Spaces: Avoidance of classical trace theorems and geometric constraints is accomplished using operator-theoretic abstract trace spaces and boundary data spaces, allowing general treatment of visco-elasticity, memory effects, or non-smooth domains (Picard et al., 2012).
• Non-Autonomous and Nonlinear Systems: Well-posedness for non-autonomous, multiplicatively perturbed, and nonlinear passive BCOS is established using Kato-type evolution families, scattering passivity, and Grönwall-type uniform bounds, including applications to time-varying coefficients and Timoshenko beams (Jacob et al., 2019, Strecker et al., 2021).

7. Applications and Illustrative Examples

• Beam and String Models: Rigorous treatments for Euler-Bernoulli and Timoshenko beams, both in port-Hamiltonian and second-order formulations, provide necessary and sufficient conditions for well-posedness, exact controllability, and implementation of structure-preserving finite-dimensional controllers (Elghazi et al., 4 Jan 2026, Elghazi et al., 28 Jan 2025, Toledo et al., 2019).
• Coupled PDEs and Parabolic-Elliptic Systems: Backstepping design and observer synthesis are achieved for coupled parabolic–elliptic systems with only partial boundary measurements, enabling output-feedback stabilization (Alalabi et al., 2023).
• Networked and Flow Systems: Abstract semigroup and control operator theory underpins the analysis and design for boundary-controlled transport and flow on graphs, including positivity and reachability properties (Engel et al., 2016).
• Nonlinear Industrial Systems: Boundary control and estimation strategies for nonlinear, nonlocal models such as under-balanced drilling are realized by combining reversed-time observer PDEs, zero-order-hold actuation, and Lyapunov stability analysis, outperforming traditional PID controllers in simulation (Strecker et al., 2021).

The breadth of BCOS encompasses robust and adaptive control, stochastic systems, fractional dynamics, distributed networks, and nonlocal/nonlinear PDEs, unified by a rigorous operator-theoretic and functional-analytic foundation. Recent research continues to refine well-posedness criteria, scalable design methodologies, and links to robust optimization and computation (Elghazi et al., 4 Jan 2026, Elghazi et al., 28 Jan 2025, Sasane, 12 Dec 2025, Phan et al., 2019).