Bilinear Optimal Control Problem

• Bilinear optimal control is a framework where system dynamics depend bilinearly on state and control variables, introducing inherent nonconvex challenges.
• The methodology leverages adjoint-based gradients, Krotov’s method, and finite element discretization to derive and ensure optimality conditions.
• Applications span chemotaxis, fluid dynamics, and quantum control, highlighting its significance in addressing complex PDE-constrained optimization.

A bilinear optimal control problem is a mathematical optimization framework where the system dynamics depend bilinearly on both the state and the control variables. Specifically, the evolution equation features terms where the state and the control appear multiplicatively—such as $N(x)u$ or $f(x,t)y(x,t)$—introducing non-affine, nonconvex dependence on the control. This structure is ubiquitous in problems where actuators modulate coefficients within physical models (e.g., reaction rates, convection velocities, damping, potentials), leading to significant theoretical and computational challenges.

1. Mathematical Formulation

A prototypical bilinear optimal control problem consists of a controlled evolution equation of the form

$$\begin{cases} \dot{x}(t) = A x(t) + N(x(t)) u(t) + B u(t) + g(t), & x(0)=x_0,\ \text{(Or for PDEs:) } \partial_t y + \mathcal{A} y + f(x) y = \dots, \end{cases}$$

where $A$, $N$, $B$ are matrices (finite dimensional) or operators (in PDEs), and $u$ is the control input. The objective is to minimize a cost functional—most often quadratic in the state and control: $$J(u) = \frac{1}{2} x(T)^\top S x(T) + \frac{1}{2} \int_0^T [x(t)^\top Q x(t) + u(t)^\top R u(t)]\,dt,$$ subject to constraints on the control (e.g., box bounds or integrability). The nonlinearity in the control enters exclusively as a bilinear term, resulting in a nonconvex optimal control problem.

In PDE settings, the canonical example is

$$\begin{aligned} \partial_t y - \Delta y + u(x,t) y &= f \quad \text{ in } \Omega\times(0,T),\ y &= 0 \quad \text{on } \partial\Omega\times(0,T),\ y(0) &= y_0, \end{aligned}$$

with a cost functional tracking the state to a desired profile or terminal target and a Tikhonov regularization on $u$ (Yahyaoui et al., 2021, Mophou et al., 2022).

2. Existence, Regularity, and Well-Posedness

Existence of optimal solutions for bilinear problems requires careful control of the nonlinear, multiplicative terms, which generally precludes reduction to convex optimization. Typical results establish that, for bounded admissible controls (e.g., $u$ in $L^\infty$, $f$ in $L^4$, or box constraints), the state map is continuous and the cost functional is lower semicontinuous, ensuring the existence of minimizers by the direct method in the calculus of variations (González et al., 2018, Fuica et al., 9 May 2024, Otarola et al., 14 Dec 2025, Mophou et al., 2022).

The regularity of the state and adjoint equations is problem-specific:

• For semilinear elliptic or parabolic PDEs, existence and regularity follow by maximal regularity theory and bootstrapping. For instance, in chemo-repulsion models, an $L^4$ regularization on the control ensures strong solutions in spaces such as $u\in L^\infty(0,T;H^1)\cap L^2(0,T;H^2)$ (González et al., 2018, Guillén-González et al., 2018, Silva et al., 2022).
• For parabolic problems with logistic or reaction terms, a priori $L^2$ and $L^{2+\epsilon}$ bounds are employed (Silva et al., 2022).

3. First-Order Necessary Conditions

Optimality systems for bilinear problems are characterized by coupling the original (forward) state equation, a backward adjoint PDE/ODE, and a nontrivial stationarity condition. The Lagrangian approach introduces adjoint variables for all PDE constraints. A representative system is:

State system (example: bilinear chemo-repulsion)

$$\begin{cases} \partial_t u - \Delta u = \nabla\cdot(u \nabla v),\ \partial_t v - \Delta v + v = u + f v,\ u(0)=u_0, \quad v(0)=v_0, \end{cases}$$

with control $f$ supported in a subdomain.

Adjoint system

$$\begin{cases} \partial_t \lambda + \Delta\lambda - \nabla\cdot(v \nabla \lambda) + \eta = -\alpha_u (u-u_d),\ \partial_t \eta + \Delta \eta - \eta - \nabla\cdot(u \nabla\lambda) + f \eta = -\alpha_v (v - v_d),\ \lambda(T) = \eta(T) = 0, \end{cases}$$

subject to suitable boundary conditions.

Optimality condition (for $f$ in $L^4$ and with cost regularization): $$\int_{Q_c} [N f^3 + v \eta](g-f) \,dx\,dt \geq 0 \quad \forall g \in F,$$ which, when unconstrained, yields the explicit control formula: $f = [-v\eta/N]^{1/3}.$ Analogous conditions are derived for other settings, where the nonconvex dependence causes stationarity conditions to be nonlinear and possibly set-valued if the admissible set features box or positivity constraints (González et al., 2018, Otarola et al., 14 Dec 2025, Fuica et al., 9 May 2024, Breiten et al., 2017, Silva et al., 2022).

4. Second-Order Optimality and Local Minimality

Due to nonconvexity, second-order conditions are essential for characterizing strict local minima. These are generally formulated in terms of the critical cone $C_{\bar u}$ of admissible perturbations and the second derivative of the reduced cost $j''(\bar u)$. Uniform positivity of $j''(\bar u)[h,h]$ over $C_{\bar u}\setminus\{0\}$ implies quadratic growth of the cost, ensuring local uniqueness and stability of the optimal control (Otarola et al., 14 Dec 2025, Fuica et al., 9 May 2024, Casas et al., 2017, Mophou et al., 2022).

A “no-gap” condition is often sought, where second-order sufficiency implies not only local optimality but also that Newton-type methods converge superlinearly (Casas et al., 2 Mar 2024, Casas et al., 2017). In bang-bang settings, such as pure tracking problems with box-constraints, it is proved that only bang-bang solutions admit quadratic growth in the $L^1$-topology (Casas et al., 2017).

5. Numerical Methods and Discretization

The nonlinearity of bilinear optimal control problems necessitates specialized algorithms.

Numerical approaches include:

• Krotov’s method, iteratively constructing monotonic sequences of improvements via Riccati-like backward and forward ODEs (Halperin, 2022).
• Gradient-based methods utilizing adjoint state computations, Newton-type iterations, and projected gradient steps (including semismooth Newton methods for nonsmooth constraints) (Casas et al., 2 Mar 2024, Otarola et al., 14 Dec 2025).
• Data-driven methods, where, under persistent excitation, trajectories and optimal policies can be synthesized entirely from a finite dataset by transforming the control problem into an equivalent nonconvex finite-dimensional program (Yuan et al., 2021).
• Finite Element discretization, with a variety of control discretizations, ranging from piecewise-constant to continuous, and a posteriori error estimators that achieve global reliability and local efficiency (Allendes et al., 29 Apr 2024, Fuica et al., 9 May 2024).
• Iterative two-point boundary value solvers, via homotopy perturbation or iterative linearization (transforming the bilinear problem into a sequence of linear-quadratic subproblems) (Ramezanpour et al., 2012, Wang et al., 2016).

For practical PDE-constrained bilinear problems, adaptive mesh refinement using residual-based error indicators sharply resolves interior or corner singularities for both the state and the control (Allendes et al., 29 Apr 2024, Fuica et al., 9 May 2024).

6. Applications and Problem Classes

Bilinear optimal control appears in a range of PDE models and application domains:

• Chemotaxis/chemo-repulsion: controlling cell aggregation or chemical spread by manipulating reaction coefficients via bilinear terms (González et al., 2018, Guillén-González et al., 2018, Silva et al., 2022).
• Fractional and nonlocal equations: optimizing bilinear controls in systems with the fractional Laplacian, subject to unique regularity and maximum principle considerations (Mophou et al., 2022).
• Stokes–Brinkman or Navier–Stokes systems: flow control by modulating local permeability or drag coefficients (Allendes et al., 29 Apr 2024).
• Maxwell’s equations: time-harmonic electromagnetic field design by optimizing spatially varying coefficients (Fuica et al., 9 May 2024).
• Semilinear elliptic PDEs with pointwise tracking: inverse design or point-matching via coefficient modulation (Otarola et al., 14 Dec 2025).
• Quantum mechanics and Schrödinger equations: external fields controlling wavefunctions through time-dependent bilinear potentials, including stochastic influences (Barbu et al., 2016).
• Ensemble control: steering an infinite family of systems parametrized by uncertainty (e.g., in NMR applications) (Wang et al., 2016).
• Monotone systems: infinite-horizon problems where optimal controls collapse to constants, reducing infinite-dimensional optimization to finite convex programs (Dhingra et al., 2016).

7. Special Topics and Current Trends

Recent research has addressed several advanced topics:

• Bang-bang optimality and singular arcs: Characterization of quadratic growth and second-order conditions for bang-bang and non-bang-bang solutions (Casas et al., 2017).
• Taylor series and generalized Lyapunov equations: Expanding the value function for feedback synthesis and model reduction in high-dimensional/infinite-dimensional settings (Breiten et al., 2017, Breiten et al., 2017).
• Boundary and coefficient control: Bilinear control acting on domain boundaries (e.g., Robin/Neumann coefficient design) or as space-dependent coefficients, with superlinear Newton convergence and precise regularity analysis (Casas et al., 2 Mar 2024, Otarola et al., 14 Dec 2025).
• A posteriori error analysis: Development of residual-based estimators for adaptive mesh refinement with theoretical guarantees, applicable in both elliptic and Maxwellian frameworks (Allendes et al., 29 Apr 2024, Fuica et al., 9 May 2024).

The diversity and depth of these investigations reflect the centrality of bilinear optimal control in both theoretical and applied PDE-constrained optimization, as well as in data-driven control, quantum systems, and engineering design.