Bilinear Approx. Controllability in Infinite Systems

• Bilinear approximate controllability is the property by which systems governed by PDEs with multiplicative (bilinear) controls can be steered arbitrarily close to any desired state.
• Explicit spectral techniques and target-killing control formulas enable precise modal decompositions and error control using static or dynamic strategies.
• Weighted Sobolev spaces, maximum principles, and geometric control arguments ensure sign-preservation and convergence in infinite-dimensional systems.

Bilinear Approximate Controllability is a foundational concept in control theory of linear and nonlinear infinite-dimensional systems, describing the property of steering the state of a system governed by a partial differential equation (PDE) with a control that acts multiplicatively on the state variable, to arbitrarily close proximity of any prescribed target state within a certain function space norm. The term “bilinear” indicates that the control mechanism interacts with the state as a product, yielding fundamentally different control structures compared to additive (affine) controls. Rigorous approximate controllability is often established in infinite-dimensional settings with spectral techniques, geometric control, and explicit control constructions.

1. Definition and General Framework

In bilinear control systems, the time evolution of the state $v(t, x)$ is controlled via a term of the form $\alpha(t, x) v(t, x)$ added to the dynamics, such as in degenerate parabolic equations: $v_t - (a(x) v_x)_x = \alpha(t, x) v$ subject to initial and boundary conditions, where $a(x)$ may vanish at the boundaries (strong degeneracy) if certain integrability holds (Cannarsa et al., 2011). The control $\alpha$ is drawn from a class of admissible functions, typically $L^\infty$ or piecewise-static functions.

Approximate bilinear controllability in $L^2(\Omega)$ (or other function spaces) is defined by the existence, for any $\varepsilon > 0$, of a control and final time $T$ such that the $L^2$-distance between the solution at time $T$ and a desired target state $v_d$ is less than $\varepsilon$: $$\| v(T, \cdot) - v_d \|_{L^2(\Omega)} < \varepsilon$$ with possible positivity or sign constraints imposed on $v_0$, $v_d$, and $v$ throughout $Q_T = (0, T) \times \Omega$.

2. Principal Theorems and Explicit Control Synthesis

Major results include constructive theorems for linear parabolic systems with bilinear control. For the strongly degenerate case, explicit control profiles are engineered as follows (Cannarsa et al., 2011):

• For a nonzero, nonnegative $v_0$ and target $v_d \geq 0$, there exists a static control $\alpha(x)$ such that, for any $\varepsilon > 0$,
  • $v(t,x)\geq 0$ for all $t,x$,
  • $$\| v(T, \cdot) - v_d \|_{L^2(\Omega)} \leq \varepsilon$$.

The construction uses the spectral characterization of the degenerate operator

$A_0 u := (a(x) u_x)_x$

and leverages the existence of an orthonormal eigenbasis $\{\phi_k\}$ with real eigenvalues $-\lambda_k$. The “target-killing” control is then

$\alpha_*(x) = -\frac{(a(x) v_d'(x))'}{v_d(x)}$

with a constant shift $\beta$ chosen to match the final time amplitude: $$\beta = \frac{1}{T} \ln \left(\frac{\|v_d\|^2}{\langle v_0, v_d \rangle}\right)$$ so that the full control is $\alpha(x) = \alpha_*(x) + \beta$. This ensures the principal mode aligns with the target, and all other modes decay exponentially, allowing error control via the spectral gap $\lambda_2-\lambda_1$.

Sign-relaxed versions are obtainable under the weaker inner product condition $\langle v_0, v_d \rangle > 0$.

3. Spectral and Functional Analytic Foundations

Accurate analysis hinges on embedding the problem in weighted Sobolev spaces, with $$H^1_a(\Omega) = \{u \in L^2(\Omega) : \sqrt{a} u_x \in L^2(\Omega)\}$$, and on compactness of embedding $H^1_a \to L^2$ given the integrability of $A(x) = \int_0^x 1/a(s) ds$.

Spectral theory of the degenerate operator ensures a discrete spectrum and orthogonal decomposition, vital for crafting modal controls.

The sign-preservation in the evolution is established via a maximum principle argument, multiplexing the equation by the negative part of the state and integrating, which implies nonnegativity for positive initial data and bounded controls.

4. Relation to Nonlinear and Ensemble Systems

Extensions to semilinear degenerate parabolic PDEs with nonlinearity $f(t,x,u)$ are available when appropriate Carathéodory and monotonicity conditions are imposed (Floridia, 2014). In these cases, piecewise-static bilinear controls can steer the system from any nonnegative initial state to any neighborhood of a nonnegative target state. The control is typically partitioned into successive phases to compress, reorient, and then rescale the state norm, with detailed energy and embedding estimates ensuring convergence.

In quantum control, bilinear approximate controllability is analyzed for ensembles of Schrödinger equations with bounded and unbounded coupling terms, spectral non-resonance, and bounded controls (Ammari et al., 2010, Chambrion, 2013, Boscain et al., 2012), yielding density of the reachable set under topological irreducibility and connectivity conditions.

5. Control Strategies: Static vs. Dynamic Bilinear Input

Static bilinear controls depend only on spatial variables and are tractable in spectral terms, while dynamic (time-dependent) bilinear controls afford temporally structured maneuvering. For strongly degenerate systems, static controls suffice for approximate controllability. For nonlinear systems or for more involved spectral landscapes, piecewise-static or fully time-dependent controls may be required, executed in concatenated segments tailored to the dynamical phases (compression, target transfer, normalization) (Floridia, 2014).

Key control formulas for static $\alpha(x)$: $$\alpha_*(x) = -\frac{(a(x) v_d'(x))'}{v_d(x)}, \quad \alpha(x) = \alpha_*(x) + \beta, \quad \beta = \frac{1}{T} \ln\left(\frac{\|v_d\|^2}{\langle v_0, v_d\rangle}\right)$$ with explicit spectral expansion of the solution: $$v(t,x) = \sum_{k=1}^\infty e^{(-\lambda_k + \beta)t} \langle v_0, \phi_k \rangle \phi_k(x)$$

6. Role of Geometry, Algebraic Saturation, and Open Directions

Approximate bilinear controllability results frequently exploit geometric control mechanisms and algebraic saturation concepts. For parabolic and dispersive systems on domains with degeneracy or symmetries, geometric criteria, weighted Sobolev embeddings, and algebraic generation of dense subspaces are critical.

Contemporary research continues to explore controllability for more intricate nonlinearities, higher-dimensional manifolds, unbounded control operators, and combinations of multiplicative and additive control. Generalization to systems with stricter constraints or physical realizability is an active theoretical trajectory.

7. Summary Table of Key Constructs

Construct	Description	Reference
Spectral target-killing control	$\alpha_*(x) = -[(a(x)v_d'(x))']/v_d(x)$	(Cannarsa et al., 2011)
Approximate controllability definition	$\|v(T)-v_d\|_{L^2} < \varepsilon$	(Cannarsa et al., 2011)
Weighted Sobolev embedding	$H^1_a(\Omega) \hookrightarrow L^2(\Omega)$	(Cannarsa et al., 2011)
Maximum principle for sign-preservation	$v_0 \ge 0$ implies $v(t,x) \ge 0$	(Cannarsa et al., 2011)
Piecewise-static control partitioning	Successive phase controls	(Floridia, 2014)

Bilinear approximate controllability thus constitutes a robust paradigm for guiding infinite-dimensional systems under multiplicative control modalities, anchored by explicit spectral constructions, weighted functional analysis, and geometric-algebraic saturation arguments.