Time-Dependent Optimal Control Problems

• Time-dependent OCPs are optimization frameworks for dynamic systems with time-varying dynamics, cost functionals, and constraints.
• They integrate differential equations, state/control constraints, and cost functionals to address non-stationarity and ensure robust feasibility.
• Advanced numerical methods such as Strang splitting, adjoint-based gradients, and SDP relaxations enable efficient solutions for these complex problems.

Time-dependent Optimal Control Problems (OCPs) involve determining control functions that steer dynamical systems—frequently governed by ordinary or partial differential equations—toward optimized performance over a finite or infinite time horizon, subject to time-varying objectives and constraints. The time dependency manifests in system dynamics, cost/loss functionals, or state/control constraints, introducing analytic and computational complexities that require specialized mathematical frameworks, optimality conditions, and numerical algorithms.

1. Mathematical Formulation and Problem Structure

Time-dependent OCPs are typically formulated as constrained optimization problems over a time interval $[0,T]$:

$$\begin{aligned} &\min_{u(\cdot)} && J(x(\cdot), u(\cdot)) = \int_0^T L(x(t), u(t), t) dt + \phi(x(T), T) \ &\text{subject to} && \dot{x}(t) = f(x(t), u(t), t),\quad x(0) = x_0 \ &&& g(x(t),u(t),t)\leq 0,\;\; h(x(t),u(t),t)=0 \ &&& x(t)\in \mathcal{X}(t),\; u(t)\in \mathcal{U}(t) \end{aligned}$$

where the system is time-varying via $f$, $L$, constraints, or sets $\mathcal{X}$ and $\mathcal{U}$.

In PDE-governed OCPs, the dynamics generalize to evolution equations such as

$$\partial_t y = \mathcal{A}(t)[y] + \mathcal{B}(t)[u] + f(t),\quad y|_{t=0} = y_0,$$

where the operator $\mathcal{A}(t)$ (e.g., elliptic, parabolic, Schrödinger-type) and data are time-dependent.

Such problems are characterized by:

• Explicit time-dependence in dynamics and cost (non-stationary systems)
• Complex target-tracking or transfer objectives
• Mixed pointwise-in-time and integral constraints, including terminal/intermediate state or control constraints

2. Optimality Conditions and Existence Theory

First-order necessary conditions for time-dependent OCPs arise from variational principles or the Pontryagin Maximum Principle (PMP). For infinite-dimensional problems (e.g., PDE contexts or coupled Schrödinger equations), the Lagrangian approach couples state and adjoint systems:

• Forward state equation (e.g., Time-Dependent Kohn-Sham model (Sprengel et al., 2017)):

$$i \partial_t \psi_j(x,t) = \left(-\Delta + V_0(x) + u(t)V_u(x) + V_\mathrm{Hxc}(x,t,\rho) \right)\psi_j(x,t)$$

• Backward adjoint equation, possibly with nonlocality or nonlinear dependence (via Fréchet derivatives):

$$i \partial_t \lambda_j(x,t) = \ldots -2\beta (\rho-\rho_d)\psi_j(x,t)$$

• Gradient (optimality) condition for the control:

$$\nu u(t) + \mu(t) = 0,\qquad \mu(t) = \text{(computed from adjoint/state coupling)}$$

Existence and differentiability of the control-to-state map can be shown under regularity and coercivity assumptions, often involving the regularization of the control in $H^1(0,T)$ for time-smoothness. Compactness theorems (Rellich–Kondrachov) are used to extract converging subsequences and establish the existence of an optimal pair $(x^*,u^*)$ (Sprengel et al., 2017).

In problems including state constraints or nonautonomous dynamics, feasibility and Lipschitz continuity require time-dependent viability conditions—i.e., at the boundary of feasible sets, the system's vector field must point into the interior suitably for all $t$ (Basco, 2023). These conditions guarantee the solvability of OCPs with time-dependent and possibly nonsmooth or unbounded constraints.

3. Numerical Discretization and Algorithms

Numerical treatment of time-dependent OCPs demands discretization schemes that account for dynamics, optimality conditions, and the necessary temporal accuracy and stability.

Splitting Schemes: For quantum systems or linear PDEs, time-splitting methods such as Strang splitting separate the evolution operator into kinetic and potential parts, each possibly integrated exactly or efficiently in transformed coordinates. For instance (Sprengel et al., 2017):

$$\begin{aligned} \psi_j' &= \exp(i \delta t \Delta)\exp(-i \frac{\delta t}{2} V(\psi(t),t))\psi_j \ \psi_j(t+\delta t) &= \exp(-i \frac{\delta t}{2} V(\psi', t+\delta t))\psi_j' \end{aligned}$$

This yields second-order temporal accuracy and (with a spectral/fourier representation for the Laplacian) high-order spatial convergence.

Nonlinear Optimization: The reduced cost functional (i.e., cost as a function of the control via the state mapping) is minimized with gradient-based methods such as nonlinear conjugate gradient (NCG), using modern update formulas (e.g., Hager–Zhang), with gradients efficiently obtained from the adjoint equation solutions.

Semi-Discrete and Lyapunov-Based Evolution: Alternative methodologies (Variation Evolving Method, or VEM (Zhang et al., 2017, Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2018)) introduce an artificial "variation time" $\tau$. The candidate solution (state, control) is evolved according to:

$$\frac{\partial y}{\partial \tau} = -K (F_y - \frac{d}{dt} F_{\dot{y}})$$

or, for optimal control,

$\frac{\partial y}{\partial \tau} = -2K\, r(t,\tau)$

where $r$ includes Hamiltonian or first-order condition terms. This turns the OCP into an Initial-Value Problem (IVP) in $\tau$, which can be solved via ODE solvers (e.g., MATLAB's ode45). The evolution is globally stable and converges monotonically to the extremal solution by construction of a Lyapunov functional.

Handling Constraints: If initial guesses do not satisfy feasibility, additional penalization terms and error-damping (MEPDE) are incorporated to drive infeasibilities in dynamics and terminal constraints to zero. For state/terminal inequality constraints, evolution steps are structured so that inactive (at the optimum) constraints become feasible in finite $\tau$ (Zhang et al., 2018).

4. Specialized Frameworks and Computational Results

Numerical frameworks must be tailored to the mathematical structure of the forward model:

• Bilinear Control in Quantum Systems: The control enters the Kohn-Sham equations via an external time-dependent potential decomposed as $V_\mathrm{ext}(x,t,u) = V_0(x) + u(t) V_u(x)$, inducing a bilinear structure exploited in both forward and adjoint equations (Sprengel et al., 2017).
• Occupation Measures and SDP Relaxations: For infinite-dimensional OCPs governed by Riesz-spectral operators, modal decomposition yields a finite set of ODE modes. The OCP is transformed into an infinite-dimensional linear program via occupation measures and then approximated by a hierarchy of moment-based semidefinite programs. Lower bounds monotonically converge toward the true cost as relaxation order increases (Magron et al., 2017).
• Adjoint-Based Lagrangian Methods for PDEs: The optimality system (state, adjoint, and gradient equations) is derived via Lagrangian multipliers and verified for convergence and accuracy using manufactured solutions (Mirzaiyan et al., 22 Oct 2025). For complex geometries (e.g., patient-specific arteries in biomedical applications), spatial and temporal discretization employs robust finite volume and implicit Euler schemes.
• Handling State Constraints and Nonautonomous Dynamics: Feasibility and regularity under time-dependent state constraints are ensured by inward-pointing conditions and uniform Lipschitz bounds on the data and constraint surfaces (Basco, 2023).
• Bang-Bang and Singular Arc Regimes: For OCPs exhibiting bang-bang structure or singular arcs, advanced smoothing (e.g., normalized $L_2$-norm function) or Integrated Residual Methods (IRM) are used. IRM minimizes integrated dynamics residuals rather than imposing collocation, suppressing high-frequency oscillations near singular arcs and enabling reliable closed-loop implementations (Wang et al., 2023, Ramesh et al., 23 Apr 2025).

5. Applications and Implications

The flexible framework for time-dependent OCPs, encompassing rigorous analysis and numerics, enables broad applications:

• Quantum Control: The Kohn-Sham/TDDFT approach enables steering multi-electron quantum systems for molecular configuration transfer, state preparation, or quantum dot manipulation. Numerical experiments demonstrate high-fidelity tracking and localization with significant reduction in cost functional (Sprengel et al., 2017).
• Process and Biomedical Systems: Time-dependent control of diffusion-dominated systems, as in drug delivery, allows for spatially localized, temporally modulated therapeutic strategies. Finite volume and adjoint-based methods adapt to dominant convection or diffusion and verify accuracy via manufactured and anatomically realistic scenarios (Mirzaiyan et al., 22 Oct 2025).
• Enforcement of State Constraints in Path Planning/Economics: In scenarios with time-varying safety or regulatory constraints, the developed viability and regularity conditions directly inform the design of robust, admissible feedback controls (Basco, 2023).
• Control of Systems with Nonlinear, Switched, or Delayed Dynamics: Modal, occupation measure, or Riccati-based solvers make these frameworks extensible to systems exhibiting fast/slow regimes, switches, or time delay, with linear complexity in horizon length for certain algorithms (Magron et al., 2017, Katayama et al., 2021, Zhang et al., 12 Apr 2024).

6. Performance, Scaling, and Implementation Considerations

The most effective schemes exhibit the following computational and practical properties:

• Second-Order Temporal Accuracy and Spectral Spatial Convergence: Demonstrated via Strang splitting for Schrödinger equations or spectral methods for linear PDEs.
• Robustness to Initial Guess and Global Feasibility Restoration: VEM-based evolution, via globally stable ODEs, enables convergence from arbitrary (even infeasible) initial guesses.
• Parallelism and Modular Decomposition: Many frameworks (e.g., modal methods, occupation measure relaxations, multi-phase NMPC) naturally parallelize, allowing competitive use in high-dimensional or long-horizon settings.
• Constraint Satisfaction and Regularization: Penalizing the $H^1$ norm of the control and integrating additional terms to drive constraint violations ensures well-posedness and convergence.
• Real-Time and Data-Driven Extensions: Data-driven reduced-order modeling (e.g., DMDc) and neural-operator surrogate approaches further accelerate simulation and enable rapid control synthesis for time-critical and feedback applications (Donadini et al., 2021, Feng et al., 17 Dec 2024).

7. Summary Table of Key Methods and Features

Methodology/Framework	System Type	Notable Features
Strang Splitting + NCG	TDKS/Schrödinger PDE	Spectral spatial accuracy, bilinear control, adjoint-based gradient, large-scale simulations (Sprengel et al., 2017)
Variation Evolving Method (VEM)	General OCPs	Lyapunov-driven gradient flow in $\tau$, turns OCP into ODE IVP, flexible handling of infeasibility (Zhang et al., 2017, Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2018)
Modal SDP Relaxation	PDEs with Riesz op.	Hierarchy of semidefinite relaxations, occupation measure lifting, convergence guarantees (Magron et al., 2017)
Adjoint-based Lagrangian	Advection-diffusion/Hemodynamics	Efficient sensitivity, complex geometry, distributed/concentrated controls (Mirzaiyan et al., 22 Oct 2025)
IRM/Integrated Residual	Problems with singular arcs	Fluctuation suppression, robust closed-loop, no a priori phase identification (Ramesh et al., 23 Apr 2025)

• (Sprengel et al., 2017) Investigation of optimal control problems governed by a time-dependent Kohn-Sham model
• (Zhang et al., 2017) A Variation Evolving Method for Optimal Control
• (Magron et al., 2017) Optimal Control of PDEs using Occupation Measures and SDP Relaxations
• (Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2018) Series on VEM and Evolution PDEs for OCPs
• (Basco, 2023) Control problems on infinite horizon subject to time-dependent pure state constraints
• (Wang et al., 2023, Ramesh et al., 23 Apr 2025) Bang-Bang Smoothing and Residual Methods
• (Mirzaiyan et al., 22 Oct 2025) On an adjoint-based numerical approach for time-dependent optimal control problems of biomedical interest

These efforts collectively advance the mathematical and numerical treatment of time-dependent OCPs, enabling high accuracy, robustness, and flexibility in quantum, biological, economic, and engineering applications.