Embedded Optimal Control Problem (EOCP)

• EOCP is a continuous relaxation of an optimal control problem that embeds discrete mode selections into continuous variables for applying classical nonlinear control methods.
• It utilizes convex embedding and concave penalty techniques to enforce bang-bang solutions, avoiding the complexity of combinatorial mode enumeration.
• Numerical methods such as direct collocation, multiple shooting, and ADMM enable scalable, real-time implementations in hybrid, power management, and PDE-constrained applications.

An Embedded Optimal Control Problem (EOCP) is a continuous relaxation of an original optimal control problem (OCP) whose dynamics or constraints involve discrete elements, typically arising from switching among multiple subsystems or operating modes. The EOCP formalism enables the application of classical nonlinear optimal control and optimization techniques to originally combinatorial or hybrid systems, by embedding the discrete selection variables into a continuous domain subject to simplex or box constraints. EOCPs have become a foundational approach for switched optimal control, hybrid system power management, elliptic PDE-constrained control with box constraints, stochastic adaptive control formulations with latent uncertainty, and real-time model predictive control in embedded settings (Sakha et al., 14 Dec 2025, Meyer et al., 2012, Allamaa et al., 2022, Song et al., 2016, Cohen et al., 2023).

1. Mathematical Formulation and Embedding Transformation

A canonical setting is the switched optimal control problem (SOCP), modeling systems governed by $N$ distinct subsystems, each activated via a binary indicator $\delta_i(t)\in\{0,1\}$, with the constraint $\sum_{i=1}^N\delta_i(t)=1$ enforcing single-mode activation. For continuous state $x(t)\in\mathbb{R}^n$, control $u(t)\in\mathcal{U}\subset\mathbb{R}^m$, and cost structure: $$\begin{array}{ll} \text{Minimize} & J = \int_{t_0}^{t_f} \ell(x(t),u(t),\delta(t),t)\,dt + \phi(x(t_f)) \ \text{subject to} & \dot{x}(t) = \sum_{i=1}^N \delta_i(t)f_i(x(t),u(t)),\, x(t_0)=x_0 \ & u(t)\in\mathcal{U},\, \delta_i(t)\in\{0,1\},\, \sum_{i=1}^N\delta_i(t)=1 \end{array}$$ (Sakha et al., 14 Dec 2025, Meyer et al., 2012).

The EOCP "embedding" replaces discrete variables $\delta_i(t)$ with continuous weights $\alpha_i(t)\in[0,1]$ over the simplex $\sum_{i=1}^N\alpha_i(t)=1$, yielding the continuous OCP: $$\begin{array}{ll} \text{Minimize} & J_e = \int_{t_0}^{t_f} \sum_{i=1}^N \alpha_i(t)\ell_i(x(t),u(t),t) dt + \phi(x(t_f)) \ \text{subject to} & \dot{x}(t) = \sum_{i=1}^N \alpha_i(t)f_i(x(t),u(t)),\, x(t_0)=x_0 \ & \sum_{i=1}^N\alpha_i(t)=1,\, \alpha_i(t)\in[0,1],\, u(t)\in\mathcal{U} \end{array}$$ This convexifies the originally combinatorial problem and enables solution by continuous optimization methods (Sakha et al., 14 Dec 2025, Meyer et al., 2012).

2. Theoretical Properties and Regularization

The EOCP preserves feasibility and, under convexity assumptions (affine $f_i$ in $u$, convex $\ell_i$ in $u$), inherits convexity in $(u(\cdot),\alpha(\cdot))$ (Meyer et al., 2012). However, the continuous optimum often yields "fractional" mode weights, i.e., $\alpha_i(t)$ in $(0,1)$ on sets of positive measure, precluding direct implementation on purely switched hardware (Sakha et al., 14 Dec 2025, Sakha et al., 9 Jan 2025).

To enforce bang-bang (pure-mode) solutions, the cost functional is augmented with a concave penalty: $$L_M(\alpha) = -\gamma \sum_{i=1}^N \zeta(\alpha_i),$$ with $\zeta(\alpha_i)$ concave, peaking at $\alpha_i=0,1$ (e.g., $\zeta(\alpha)=\alpha(1-\alpha)$) (Sakha et al., 14 Dec 2025, Sakha et al., 9 Jan 2025). This leads to the Modified EOCP (MEOCP): $$J_m = \int_{t_0}^{t_f} \left[L(x,u,\alpha,t) + L_M(\alpha(t))\right] dt + \phi(x(t_f))$$ Under Pontryagin’s Principle, the minimizer in $\alpha$ for the concave-augmented Hamiltonian is always attained at a simplex vertex—yielding $\alpha_i\in\{0,1\}$ a.e. and thus feasibility for the original SOCP (Sakha et al., 14 Dec 2025, Sakha et al., 9 Jan 2025).

3. Numerical Solution Methods

EOCP and MEOCP are transcribed to finite-dimensional nonlinear programs via either direct collocation (orthogonal or polynomial), direct multiple-shooting, or finite element/finite difference discretizations.

Direct Collocation: Introduce variables $\{x_k, u_k, \alpha_k\}$ at collocation nodes, enforce system dynamics via collocation defects, impose simplex and box constraints on $\alpha$, and solve the resulting smooth NLP (e.g., IPOPT, SQP, active-set QP) (Sakha et al., 14 Dec 2025, Allamaa et al., 2022, Uthaichana et al., 2018).

Orthogonal Collocation with Safety Envelopes: Leverages Legendre polynomial state parameterization and a Bernstein-based envelope to enforce state and input constraints over the trajectory, guaranteeing continuous feasibility and high-order convergence. The envelope tightness provides minimal conservatism, essential for embedded and real-time contexts (Allamaa et al., 2022).

ADMM and PDAS for Elliptic EOCPs: For elliptic PDE-constrained OCPs with pointwise box constraints ("EOCP"s in the literature of control-constrained PDEs), finite element discretization is combined with inexact heterogeneous ADMM for fast, mesh-independent convergence, optionally refined with PDAS semismooth Newton methods (Song et al., 2016).

Discretization Method	Key Features	Applications
Direct collocation (LGL, PS)	Spectral convergence, small NLP size	NMPC, hybrid powertrain, AVP
Multiple shooting	Parallelizable, easy state constraint	Real-time NMPC, embedded
FE/FV + ADMM	Mesh independence, large-scale PDEs	Elliptic/Parabolic EOCPs

4. Applications and Extensions

EOCPs have broad applicability across domains involving switching, hybrid, or constrained systems:

• Powertrain control: Embedded EOCP for bi-modal hybrid electric vehicle management, with direct collocation and receding-horizon NMPC, supports real-time fuel/battery/track optimization without resorting to MIP (Uthaichana et al., 2018).
• Switched systems with dwell-time: MEOCP with additional filtering layer using mode insertion gradients constrains minimal switching intervals for hardware longevity or actuator protection (Sakha et al., 9 Jan 2025).
• PDE-constrained control: EOCP refers also to control of distributed parameter systems with box constraints, solved efficiently via two-phase ADMM/PDAS methods (Song et al., 2016).
• Safety-critical NMPC: Bernstein-augmented collocation enforces envelope-based continuous constraints, crucial for autonomous driving and embedded systems (Allamaa et al., 2022).
• Stochastic/Adaptive control: Markovian embedding of non-Markovian control into high-dimensional EOCP yields rigorous HJB characterization and $\varepsilon$-optimality for dual-adaptive learning policies under filter-dependent information (Cohen et al., 2023).

5. Connections to Alternative Approaches

EOCP offers several computational and theoretical advantages over classical methods such as mixed-integer programming (MIP), multi-parametric programming (MPP), cyclic gradient-based algorithms, and the hybrid minimum principle:

• No combinatorial mode enumeration: Relaxes mode scheduling to continuous variables, eliminating combinatorial explosion (Sakha et al., 14 Dec 2025, Meyer et al., 2012).
• Scalability: Polynomial in state/mode/grid variables, tractable for moderate dimensions; MIP and MPP scale exponentially (Meyer et al., 2012).
• Solver compatibility: Enables off-the-shelf NLP solvers, CasADi/Acados/GPOPS-II/Matlab fmincon, real-time embedded code, and warm-start MPC (Sakha et al., 14 Dec 2025, Verschueren et al., 2019, Uthaichana et al., 2018).
• Guaranteed feasibility: Concave penalties guarantee bang-bang optimality, sidestepping chattering and ensuring implementable mode schedules (Sakha et al., 14 Dec 2025).
• Analogous performance: Empirically, embedded solutions match or outperform MIP/MPP/gradient-based methods in cost and runtime for a range of benchmarks (Meyer et al., 2012).

6. Practical Implementation and Limitations

Implementation involves grid and regularization parameter tuning, choice of envelope tightness (if relevant), and the possible need for post-processing to recover strict mode sequences or to enforce secondary constraints (e.g., dwell-time). For certain hard constraints, filtering or projection steps are required, which may disturb global optimality but can be offset by local re-optimization (Sakha et al., 9 Jan 2025, Uthaichana et al., 2018).

EOCP does not require prior knowledge of the switch sequence and accommodates both controlled and autonomous switching. However, for highly discontinuous solutions or pure state/junction constraints, finer discretization or auxiliary chattering constructions may be required (Meyer et al., 2012, Sakha et al., 14 Dec 2025). For high-dimensional PDEs, efficient ADMM/PDAS extensions offer mesh-independent scalability (Song et al., 2016). Oscillatory "fractional" solutions on singular arcs may arise, but are suppressed by the concave penalty construction (Sakha et al., 14 Dec 2025).

7. Outlook and Impact

Embedded optimal control is the unique formalism allowing efficient, provably correct, and scalable optimal synthesis for switched, hybrid, and constraint-embedded systems in both deterministic and stochastic/adaptive contexts (Sakha et al., 14 Dec 2025, Cohen et al., 2023). The method’s integration with advanced discretization (spectral, FE, collocation), envelope methods for continuous constraint satisfaction, and its favorable computational profile across diverse engineering domains substantiate its status as the dominant approach for high-performance, real-time optimal control in embedded and safety-critical applications. Empirical results consistently demonstrate reductions in solution times (up to 9x in NMPC (Allamaa et al., 2022)), lower or matching optimal costs, and avoidance of intractability issues endemic to combinatorial approaches (Meyer et al., 2012, Allamaa et al., 2022, Sakha et al., 14 Dec 2025).

EOCP’s compatibility with modular toolchains (acados, CasADi, GPOPS-II), theoretical guarantees (bang-bang optimality, convexity under mild conditions), and extensibility to stochastic and learning-based control render it indispensable in contemporary optimal control research and embedded control deployment (Verschueren et al., 2019, Cohen et al., 2023).