Switching-Time Optimal Control Problem

• The topic is defined as an optimal control framework where both control inputs and the timing of discrete switches are jointly optimized.
• It introduces techniques like time rescaling and embedding relaxations that transform discrete switching decisions into a continuous optimization problem.
• The analysis presents optimality conditions and numerical methods—including gradient-based and sequential linear-quadratic approaches—addressing dwell-time constraints and application challenges.

A switching-time optimal control problem refers to the class of optimal control problems where the timing of discrete switches between system modes, dynamic laws, or constraint sets is included as an optimization variable alongside the control itself. Such problems occur in hybrid and switched systems, diffusion-reaction and parabolic PDEs with regime changes, nonlinear and linear ODEs, stochastic delay systems, and practical engineering contexts involving constraints on switching frequency or dwell time. Formally, the system’s physical behavior, cost criteria, and admissibility depend not only on the continuous control signal $u(\cdot)$ but also on a discrete set of switching times $\tau = \{\tau_1,\ldots,\tau_N\}$ at which the system’s dynamics, control bounds, or cost functions change.

1. Mathematical Formulation and Problem Statements

Let $\Omega\subset\R^n$ be a smooth bounded domain, and $T>0$ the time horizon. The general switching-time optimal control problem for a system governed by semilinear parabolic PDEs takes the form (Court et al., 2016): $$\begin{gathered} \min_{u,\tau} J(u,\tau) = \sum_{k=1}^N \int_{\tau_{k-1}}^{\tau_k} \int_{\Omega} L_k(y(t,x),u(t,x))\,dx\,dt + \int_{\Omega}\Phi(y(T,x))\,dx, \ \text{subject to} \;\;\; \partial_t y(t,x) - \Delta y(t,x) = f_k(y(t,x),u(t,x)), \; \forall\, t\in (\tau_{k-1},\tau_k),\; k=1,\ldots,N, \ u(t,x)\in [u^-_k(x),u^+_k(x)],\quad y(0,x) = y_0(x), \quad y(t,x) = 0\text{ on }\partial\Omega. \end{gathered}$$ Here, $\{\tau_k\}$ are the sequence of switching times at which the system’s coefficients, cost, and control constraints change. The admissible control-set $V_{ad}$ enforces $u$ and $\tau$ constraints.

Hybrid switched ODE systems and nonlinear switched systems are similarly formulated as

$$\dot{x}(t) = f_{\sigma_i}(x(t),u(t)),\quad t\in [\tau_i,\tau_{i+1}),\quad \sigma = (\sigma_0,\ldots,\sigma_{M-1}),\quad \tau = \{\tau_1,\ldots,\tau_{M-1}\}.$$

The cost typically aggregates over intervals between switches, and constraints (e.g., dwell-time, sequence feasibility, endpoint matching) may further restrict scheduling (Farshidian et al., 2016, Abbasi-Esfeden et al., 9 Jan 2025).

2. Time Domain Reformulation and Embedding Strategies

To facilitate analysis and numerical optimization, problems are reformulated by smoothing discontinuities across switching times or converting combinatorial scheduling to continuous relaxation.

Time-Rescaling: By introducing a rescaled reference time $s\in[0,N]$ mapped to $t\in[0,T]$ via partition-dependent $\pi(s;\tau)$, the original problem is cast onto a fixed domain $s$ with piecewise dynamics (Court et al., 2016). Controls and states are correspondingly pulled back, and constraints are encoded as functions of $s$.

Embedding-Based Relaxation: Discrete mode signals are replaced by continuous proxies $v(t)\in[0,1]$ or $v(t)\in\Delta^{M}$ (the simplex over $M$ modes), facilitating the use of gradient-based optimization. Auxiliary concave penalty terms (e.g., $L_{dwell}(v)=b\,(v-v^2)$) enforce bang-bang structure in the relaxed solution, which encourages valid switching patterns in the original discrete system (Abudia et al., 2020, Sakha et al., 9 Jan 2025).

For discrete-time switched linear systems, block-sparsity inducing relaxations allow solving for auxiliary variables enforcing single-mode activation per time-step via group $\ell_1/\ell_2$ penalties (Kreiss et al., 2017).

3. Necessary and Sufficient Optimality Conditions

First-order necessary conditions encompass adjoint (costate) equations, Hamiltonian stationarity, and transversality with respect to the switching times:

• Adjoint Equation for Semilinear Parabolic PDEs: On sub-intervals $(\tau_{k-1},\tau_k)$,

$$-\partial_s p(s) - \dot{\pi}(s;\tau) [\Delta p(s) + D_y f_k(y(s),u(s))^\top p(s)] = \dot{\pi}(s;\tau) D_y L_k(y(s),u(s)),$$

with jump/transversality conditions at switch points and boundary/terminal conditions (Court et al., 2016).

• Pontryagin Maximum Principle (PMP): The optimal control $u^*(s,x)$ at each $s$ is determined by \begin{align*} & u^*\in(u^-_k,u^+_k) \implies D_u H_k(y^(s),u^(s),p^*(s))=0, \ & u^* = u^-_k \implies D_u H_k(y^,u^,p^*)\ge 0, \ & u^* = u^+_k \implies D_u H_k(y^,u^,p^*)\le 0, \end{align*} and the Weierstrass–Euler conditions for switching times,

$$\frac{\partial J}{\partial \tau_k} = H_k(y^*,u^*,p^*)(\tau_k^-) - H_{k+1}(y^*,u^*,p^*)(\tau_k^+) = 0.$$

Second-order sufficient conditions are established via coercivity of the second derivative of the cost functional on the critical cone of permissible variations in control and switching times.

4. Switching Structure and Regularity

The regularity and structure of optimal controls depend on underlying system geometry and Lie bracket conditions. For control-affine systems in $\mathbb{R}^n$ with $k=n-1$ controls, bracket tests provide explicit local characterizations:

• Single Switch / Smoothness: If the bracket combination $a(q)\notin A(q) S^{n-2}$ at $q$, every optimal trajectory in a neighborhood is piecewise smooth with at most one switching point. Singular arcs are absent; switchings are unique and isolated (Agrachev et al., 2016, Agrachev et al., 2016).
• Explicit Jump Formula: Near a switching time $\tau$ (isolated zero of switching functions), post-/pre-switch controls satisfy

$$u(\tau^{\pm}) = [\pm d I + H_{IJ}]^{-1} H_{0I}, \quad d>0,$$

with $d$ determined from local bracket data.

For parabolic and PDE systems, analytic continuation and unique continuation properties (e.g., vanishing of the adjoint switching function) impose finite bounds on the number of switchings per interval and enforce bang-bang transitions (Qin et al., 2020).

In generic linear time-optimal problems with purely imaginary spectra, the number of switchings $N(T)$ grows linearly with time horizon $T$,

$N(T) \ge \frac{|\Omega|}{\pi} T + o(T),$

where $\Omega$ is the mean motion of the associated complex switching function (Dalin et al., 19 Feb 2025).

5. Numerical Solution Algorithms

A broad suite of numerical methods exists for solving switching-time optimal control problems:

• Gradient-Based Optimization: Hybrid systems often employ Barzilai–Borwein or quasi-Newton schemes for joint optimization over control functions and switching times (Court et al., 2016, Aghaee et al., 2020).
• Sequential Linear-Quadratic (SLQ) Approach: Two-stage SLQ algorithms alternate between control policy improvement (via Riccati differential equations linearized around trajectories) and gradient descent over switching times using backward sensitivity equations for the value function (Farshidian et al., 2016). Gradient $\frac{\partial J}{\partial \tau_j}$ is accessible via sensitivity ODE solves.
• Riccati-Recursion for Multiple Shooting: Large-scale nonlinear switched systems admit efficient Riccati-style Newton solvers for simultaneous control and switching time optimization, scaling linearly in discretization stages and exploiting block sparsity in KKT matrices (Katayama et al., 2021).
• Direct Collocation / Nonlinear Programming: Embedded variable relaxations allow direct collocation methods with mesh refinement, enabling SQP or interior-point solvers to handle both continuous and relaxed discrete decisions (Abudia et al., 2020, Abbasi-Esfeden et al., 9 Jan 2025).
• Schedule-Space Armijo Gradient Descent: Modes and switching times are updated directly in schedule space using Gateaux derivatives ("insertion gradients"), with convergence guarantees in continuous time (Wardi et al., 2011).
• Convex Block-Sparsity Relaxations (Discrete-Time): Block $\ell_1/\ell_2$ penalties over auxiliary variables reduce search complexity from exponential in mode sequence length to polynomial time convex optimization (Kreiss et al., 2017).
• Mixed-Integer Nonlinear Programming (MINLP): Dwell-time and sequence constraints can be efficiently encoded with a fixed number of binaries by restricting to subsequences of a master schedule (Abbasi-Esfeden et al., 9 Jan 2025). Iterative heuristics (ISTO) with complementarity penalties ensure constraint satisfaction and rapid convergence.
• Bang-Bang/Singular Switch Optimization: Problems with known or finite switching/singular patterns (e.g., minimum-time/fuel LTI) can be reduced to finite-dimensional static polynomial or moment-SOS programs for the optimal switching times (Sarkar et al., 2021).

6. Dwell-Time Constraints and Regulatory Penalties

Dwell-time constraints (minimum period between switches) are crucial for modeling regulatory, physical, or operational requirements. Two families of enforcement strategies are prominent:

• Auxiliary Penalty Functions: Concave penalties on intermediate embedded mode values (e.g., $L_{dwell}(v)$) are heuristically tuned to reduce switching rates; increasing the penalty weight increases mean dwell times but does not enforce strict minimums (Abudia et al., 2020).
• Explicit Constraints via Filtering: After bang-bang relaxation, explicit filtering algorithms examine the switching schedule and remove or merge switches violating dwell-time constraints, using insertion gradients to select the least-damaging replacements (Sakha et al., 9 Jan 2025).
• MINLP Direct Encoding: Dwell-time constraints can be embedded as linear inequalities $w_k\ge\Delta_{min}$ in a master-sequence MINLP, decoupling difficulty from grid resolution (Abbasi-Esfeden et al., 9 Jan 2025).

Numerical studies illustrate the trade-off between cost performance and switching frequency, with dwell-time enforcement typically increasing the cost but guaranteeing feasibility.

7. Applications and Representative Examples

Switching-time optimal control problems are pervasive in:

• Diffusion-reaction and population dynamics: As in the Lotka–Volterra controlled prey-predator system in (Court et al., 2016), optimal intervention timing trades off local/maximal species population with long-term targets.
• Quadrupedal robot CoM control: Sequential linear-quadratic methods manage phase compression in dynamic locomotion planning (Farshidian et al., 2016).
• Hydro-power plant scheduling: Stochastic impulse switching in the presence of hydrological coupling and delay maximizes long-term revenue (Perninge, 2019).
• Quantum control: Binary quantum gates and switch-duration optimization yield high-fidelity quantum transformations (Fei et al., 2023).
• Portfolio selection and retirement timing: Piecewise job-choice and discretionary stopping points optimize consumption and investment under borrowing constraints (Jeon et al., 2021).
• Bang-bang/singular time and fuel minimization: For LTI systems, optimal bang-off-bang structures and timing sequences are globally constructed using generalized moment problems (Sarkar et al., 2021).

8. Key Theoretical Insights and Structural Phenomena

1. Hamiltonian Constancy: For autonomous systems (time-independent dynamics and costs), the Hamiltonian evaluated along the optimal trajectory is piecewise constant, including across switching instants (Court et al., 2016).
2. Switching-Time Derivative Formulas: The derivative of the objective with respect to switching times is computable via jump differences in Hamiltonians at switch points or via adjoint matching conditions (Aghaee et al., 2020).
3. Regularity and Boundedness: In parabolic and control-affine systems, unique continuation and bracket conditions restrict switching points to be isolated and finite within any bounded interval (Qin et al., 2020, Agrachev et al., 2016).
4. Complexity and Scalability: Advanced algorithms (embedded relaxations, master-sequence MINLPs, Riccati recursions) break combinatorial barriers, enabling scalability to high-dimensional, grid-refined, or large-horizon switching problems.
5. Sensitivity and Verification: The Hamilton–Jacobi-Bellman approach connects flow derivatives w.r.t. initial conditions to sensitivity of switching times, enabling rigorous verification of optimality in bang-bang solutions (Riquelme, 2020).

9. Limitations, Open Challenges, and Future Directions

• Precise analytical relations between dwell-penalty weights and minimum dwells remain open.
• Scalability for general nonlinear dynamics, combinatorial sequence optimization in high-mode-count settings.
• Extension and robustness of schedule-space algorithms to stochastic dynamics, pathwise uncertainty, or impulse switching with continuous delays.
• Integration of learning-based or data-driven co-design of switching sequences with classical dynamic programming frameworks.

Switching-time optimal control is central in hybrid systems, providing rigorous structure and efficient algorithms for joint continuous-discrete optimality, and underpins diverse applications in engineering, physics, economics, and stochastic control theory.