Terminal Constraints in Optimal Control

• Terminal constraints are endpoint conditions, defined as exact equalities or inequalities, that ensure feasible and safe solutions in optimal control and MPC.
• They can be implemented as hard constraints for strict endpoint adherence or as soft penalties to promote convergence in stochastic and PDE control settings.
• Specialized numerical methods, such as the Variation Evolving Method (VEM) and its extensions, restore feasibility and shape solver structures by handling both deterministic and stochastic endpoint requirements.

Searching arXiv for recent and foundational papers on terminal constraints in optimal control, MPC, stochastic control, and related formulations. Terminal constraints are endpoint conditions imposed at the final stage of a control, planning, or optimization problem. In the cited literature, they appear as exact terminal equalities such as $x(N+1)=\xi$, terminal inequalities such as $g_I(x(t_f),t_f)\le 0$, terminal set membership conditions such as $h(x_{t+N|t})\ge 0$, terminal constraints in law of the form $\Psi(\mathcal L(X_T))\le 0$, and strict stochastic constraints such as $X_T=0$ or $X_T\in C$ for a prescribed random subspace (Xu et al., 2023). Across deterministic optimal control, model predictive control, stochastic linear-quadratic control, mean field control, and PDE control, terminal constraints serve different but related roles: exact endpoint feasibility, recursive feasibility, recoverability, safety certification, convergence regularization, and structural characterization of admissible optimal solutions (Zhang et al., 2018).

1. Deterministic optimal control with terminal equality and inequality constraints

A canonical deterministic formulation is the Bolza optimal control problem

$$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$

subject to

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$

with free terminal time $t_f$, terminal equality constraint

$g(x(t_f),t_f)=0,$

or, in the more general case,

$g_I(x(t_f),t_f)\le 0$0

(Zhang et al., 2018). In this setting, terminal equality constraints are exact endpoint conditions, whereas terminal inequality constraints may be active or inactive at the optimum. The active-set distinction is central: an inequality is active when the terminal value is zero and inactive when it is strictly interior (Zhang et al., 2018).

The Variation Evolving Method (VEM) treats the optimal solution as the equilibrium of an infinite-dimensional dynamical system evolving in a fictitious variation time $g_I(x(t_f),t_f)\le 0$1. In the terminal-equality case, admissible variations must satisfy

$g_I(x(t_f),t_f)\le 0$2

which is the feasibility condition in variation time: the evolution must stay on the terminal-constraint manifold (Zhang et al., 2018). The resulting evolution law preserves feasibility, decreases the cost monotonically, and converges asymptotically to stationary conditions equivalent to Pontryagin’s necessary conditions (Zhang et al., 2018).

For terminal inequalities, the VEM treatment is more delicate. The problem includes both equality and inequality endpoint conditions,

$g_I(x(t_f),t_f)\le 0$3

and only the correct subset of terminal inequalities should be enforced as equality-type conditions in the variation subproblem (Zhang et al., 2018). The paper introduces the active index set $g_I(x(t_f),t_f)\le 0$4 and the refined set $g_I(x(t_f),t_f)\le 0$5, the latter selecting those active inequalities whose variation constraints are themselves active in the feasibility-preserving evolution optimization problem (Zhang et al., 2018). This distinction prevents saturated but ultimately inactive inequalities from being frozen unnecessarily.

The deterministic VEM literature also develops explicit costate-free optimality conditions. In the terminal-equality case they are

$g_I(x(t_f),t_f)\le 0$6

and

$g_I(x(t_f),t_f)\le 0$7

where $g_I(x(t_f),t_f)\le 0$8 is obtained algebraically from

$g_I(x(t_f),t_f)\le 0$9

rather than by solving an adjoint two-point boundary value problem (Zhang et al., 2018). In the terminal-inequality extension, the same structure persists with equality multipliers $h(x_{t+N|t})\ge 0$0 and inequality multipliers $h(x_{t+N|t})\ge 0$1, and the KKT conditions

$h(x_{t+N|t})\ge 0$2

appear explicitly (Zhang et al., 2018).

A later modification, the Modified Evolution Partial Differential Equation (MEPDE), extends VEM to the infeasible solution domain. It allows initial guesses that violate the dynamics, initial condition, or terminal constraints, and uses first-order stable dynamics to eliminate these infeasibilities (Zhang et al., 2018). A distinctive result is that violated terminal inequality constraints that are inactive for the optimal solution enter the feasible domain in finite variation time, whereas active terminal inequalities are achieved asymptotically (Zhang et al., 2018). This separates feasibility restoration from asymptotic optimality in a precise way.

2. Terminal sets, terminal admissible regions, and terminal regularization in model predictive control

In model predictive control, terminal constraints often appear as terminal sets rather than exact terminal state equalities. One safety-critical formulation uses a discrete-time nonlinear system

$h(x_{t+N|t})\ge 0$3

with a signed-distance safety constraint $h(x_{t+N|t})\ge 0$4 imposed along the horizon and a terminal control barrier function constraint

$h(x_{t+N|t})\ge 0$5

imposed only at the endpoint (Dokania et al., 7 May 2026). Here the terminal set is the CBF-safe set

$h(x_{t+N|t})\ge 0$6

which is controlled invariant, while the stagewise constraint uses the true geometric unsafe set encoded by $h(x_{t+N|t})\ge 0$7 (Dokania et al., 7 May 2026). This construction separates actual collision avoidance from terminal recoverability. The terminal CBF condition guarantees recursive feasibility and safety through recoverability, while reducing conservatism relative to stage-wise CBF enforcement (Dokania et al., 7 May 2026).

The same work formalizes horizon-dependent reachable sets. For the proposed MPC-MCI scheme, the one-step reachable sets satisfy

$h(x_{t+N|t})\ge 0$8

whereas for the compared NMPC-DCBF formulation the reachable set is effectively horizon-invariant: $h(x_{t+N|t})\ge 0$9 (Dokania et al., 7 May 2026). The constructive recursive-feasibility proof also yields a warm-start strategy: shift the previous control sequence and append a recovery action from the invariant terminal set (Dokania et al., 7 May 2026). This suggests that, in safety-critical MPC, the terminal constraint can function as an invariant recovery anchor rather than merely a stabilizing endpoint condition.

A different MPC use of terminal constraints appears in image-based visual servoing of UAVs. There, the error dynamics are

$\Psi(\mathcal L(X_T))\le 0$0

and the optimization includes stage cost, terminal cost, state and input bounds, and the terminal-state constraint

$\Psi(\mathcal L(X_T))\le 0$1

(Wang et al., 21 May 2026). The terminal admissible region is therefore

$\Psi(\mathcal L(X_T))\le 0$2

This terminal set is described as a near-target admissible terminal region and a convergence regularizer, but the paper does not provide a theorem on recursive feasibility, asymptotic stability, invariant-set construction, or Lyapunov decrease (Wang et al., 21 May 2026). The formulation is thus an empirically validated terminal-constrained MPC design rather than a formally developed stability theory.

Learning-based MPC introduces another endpoint interpretation. In collision avoidance with dynamic obstacles, the terminal constraint is a learned approximation of the Hamilton–Jacobi maximal safe set at time $\Psi(\mathcal L(X_T))\le 0$3: $\Psi(\mathcal L(X_T))\le 0$4 with

$\Psi(\mathcal L(X_T))\le 0$5

(Derajić et al., 5 Aug 2025). Because $\Psi(\mathcal L(X_T))\le 0$6, one has

$\Psi(\mathcal L(X_T))\le 0$7

so every state unsafe according to the signed distance function baseline remains unsafe under the learned terminal approximation (Derajić et al., 5 Aug 2025). The resulting terminal set is time-varying, environment-conditioned, and updated online from local obstacle predictions. This shifts terminal constraints away from fixed invariant geometry toward learned safe-set surrogates.

A further MPC-related variation is the generalized terminal constraint in data-driven economic MPC for unknown LTI systems. Instead of forcing the terminal prediction to a fixed optimal equilibrium, the method requires that the final $\Psi(\mathcal L(X_T))\le 0$8 input-output samples form a constant equilibrium segment: $\Psi(\mathcal L(X_T))\le 0$9 (Xie et al., 2022). The equilibrium $X_T=0$0 is an optimization variable. This enlarges the feasible region and avoids prior knowledge of the optimal equilibrium, while still supporting asymptotic average performance guarantees (Xie et al., 2022).

3. Hard, soft, and generalized terminal constraints

A recurring theme in the cited literature is the distinction between hard terminal constraints and soft terminal penalties. In deterministic VEM, the terminal equality

$X_T=0$1

is hard: all iterates must remain on the terminal-constraint manifold once feasibility is imposed (Zhang et al., 2018). In contrast, the heat-equation control paper replaces the exact terminal condition

$X_T=0$2

by the penalized functional

$X_T=0$3

and proves convergence of the penalized controls and terminal states to the exact constrained solution as $X_T=0$4 (Kwon, 13 May 2026).

For the one-dimensional heat equation, the hard-constrained minimum-energy solution exists uniquely when the target is admissible, and the soft-constrained modal minimizers are explicit: $X_T=0$5 (Kwon, 13 May 2026). The convergence is quantitative: $X_T=0$6 and the paper gives rates $X_T=0$7 and the sharp $X_T=0$8 rate under stronger spectral summability conditions (Kwon, 13 May 2026). This makes the soft-to-hard transition fully explicit rather than purely asymptotic.

The same hard-versus-soft distinction appears in stochastic linear-quadratic control with random terminal subspace constraints. There the hard constraint is

$X_T=0$9

with $X_T\in C$0 a prescribed random linear subspace (Ackermann et al., 7 Jan 2026). It is approximated by penalized terminal costs

$X_T\in C$1

which vanish on $X_T\in C$2 and grow like $X_T\in C$3 in directions orthogonal to $X_T\in C$4 (Ackermann et al., 7 Jan 2026). The singular limit yields a Riccati BSDE with symbolic terminal condition

$X_T\in C$5

made rigorous via quadratic-form blow-up outside $X_T\in C$6 and domination by $X_T\in C$7 on $X_T\in C$8 (Ackermann et al., 7 Jan 2026).

A related mean field control note uses a particle formulation where exact terminal measure matching is not imposed. Instead, the empirical terminal measure is matched to a target empirical measure through a Gaussian blob penalty

$X_T\in C$9

(Craig et al., 2024). The terminal condition is therefore a soft nonlocal approximation of a target-measure constraint rather than a hard equality of measures (Craig et al., 2024). This suggests a broader interpretation of terminal constraints as endpoint distribution matching under regularization.

4. Stochastic terminal constraints, singular terminal conditions, and terminal constraints in law

In stochastic control, terminal constraints can be random, set-valued, or distributional. One scalar stochastic LQ problem imposes a partial terminal constraint through a random terminal penalty parameter $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$0: $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$1 while on $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$2 terminal deviation is penalized quadratically in the cost

$$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$3

(Bank et al., 2016). The paper shows that the classical coupled Riccati/linear-BSDE method breaks down under such partial terminal constraints and replaces it by auxiliary problems indexed by supersolutions of a singular backward stochastic Riccati equation (Bank et al., 2016). The strict terminal event $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$4 makes the endpoint condition both random and singular.

The multidimensional stochastic LQ paper with random terminal subspace constraints treats a different hard stochastic endpoint condition: $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$5 where $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$6 is a random linear subspace of $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$7 (Ackermann et al., 7 Jan 2026). Encoding $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$8 for a bounded random $$J=\varphi(x(t_f),t_f)+\int_{t_0}^{t_f} L(x(t),u(t),t)\,dt$$9 transforms the hard constraint into a singular Riccati BSDE problem. The minimal supersolution $\dot x=f(x,u,t), \qquad x(t_0)=x_0,$0 characterizes both the value function

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$1

and the optimal feedback

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$2

(Ackermann et al., 7 Jan 2026). The blow-up of $\dot x=f(x,u,t), \qquad x(t_0)=x_0,$3 near $\dot x=f(x,u,t), \qquad x(t_0)=x_0,$4 is not incidental; it is the analytic expression of the exact terminal restriction.

Terminal constraints can also be imposed on the law of the terminal state rather than on individual sample paths. A controlled diffusion

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$5

is subject to the hard terminal law constraint

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$6

(Daudin, 2020). Using convex duality, the paper derives a coupled optimality system consisting of a forward Fokker–Planck equation and a backward HJB equation with terminal condition

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$7

together with complementarity

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$8

(Daudin, 2020). In this formulation, the terminal constraint acts through a scalar multiplier at the level of measures rather than trajectories.

Extended mean field games with common noise and strict terminal state constraint provide yet another stochastic endpoint regime. The admissible set is

$\dot x=f(x,u,t), \qquad x(t_0)=x_0,$9

so the terminal constraint is almost sure and exact (Hua et al., 9 Jun 2025). The constrained problem is solved by penalization with

$t_f$0

followed by the limit $t_f$1, yielding a new type of coupled conditional mean field FBSDE with a free backward part (Hua et al., 9 Jun 2025). The limiting state satisfies

$t_f$2

and the singular behavior of the decoupling field near terminal time enforces exact liquidation (Hua et al., 9 Jun 2025).

5. Numerical computation and algorithmic structure under terminal constraints

Terminal constraints substantially shape algorithm design. In compact VEM for terminally constrained OCPs, the EPDE is semi-discretized in physical time, converting the infinite-dimensional evolution into a finite-dimensional IVP in variation time that can be integrated with standard ODE solvers (Zhang et al., 2018). The method requires an initial feasible trajectory-control pair satisfying the dynamics, initial condition, and terminal constraint, which the paper explicitly notes as a limitation (Zhang et al., 2018).

The MEPDE removes that limitation by introducing error dynamics for the dynamics defect, the initial state defect, and the terminal constraint residuals (Zhang et al., 2018). This suggests a two-layer numerical interpretation: feasibility restoration and optimality descent proceed simultaneously in $t_f$3. The terminal inequality case additionally motivates a numerical soft barrier

$t_f$4

to damp the numerical error caused by sudden activation of terminal inequalities (Zhang et al., 2018). The paper emphasizes that this is a stabilizing correction rather than a logarithmic interior-point barrier (Zhang et al., 2018).

For exact terminal state constraints in unknown discrete-time LQ control, the Q-learning formulation incorporates the terminal equality by introducing a Lagrange multiplier $t_f$5 and learning a Q-function quadratic in the augmented variable $t_f$6 (Xu et al., 2023). The optimal controller has the affine form

$t_f$7

where $t_f$8 solves

$t_f$9

(Xu et al., 2023). Here the terminal equality is enforced exactly, not approximately, and the extra feedforward term $g(x(t_f),t_f)=0,$0 is the direct algebraic imprint of the endpoint condition (Xu et al., 2023).

Terminal constraints also alter solver structure in linear MPC. For a terminal ellipsoidal constraint

$g(x(t_f),t_f)=0,$1

a sparse ADMM solver imposes the consensus condition

$g(x(t_f),t_f)=0,$2

instead of $g(x(t_f),t_f)=0,$3, so that the terminal subproblem becomes a weighted projection onto the ellipsoid (Krupa et al., 2021). The projection has the explicit form

$g(x(t_f),t_f)=0,$4

as derived in the paper’s appendix (Krupa et al., 2021). This is a solver-level example of how terminal constraint geometry can dictate the algebraic structure of the numerical method.

6. Interpretation, roles, and recurring distinctions

Several distinctions recur across the literature.

Hard versus soft terminal conditions: Hard constraints appear as $g(x(t_f),t_f)=0,$5, $g(x(t_f),t_f)=0,$6, $g(x(t_f),t_f)=0,$7, or $g(x(t_f),t_f)=0,$8. Soft terminal conditions appear as quadratic penalties such as $g(x(t_f),t_f)=0,$9 or $g_I(x(t_f),t_f)\le 0$00 (Kwon, 13 May 2026). The cited papers repeatedly use soft penalties as approximations to hard constraints, but only some provide quantitative or structural convergence results (Kwon, 13 May 2026).

Equality versus inequality terminal constraints: Equality constraints define exact terminal manifolds. Inequality constraints require activity analysis, KKT multipliers, and often active-set logic (Zhang et al., 2018). The deterministic VEM papers show that active terminal inequalities behave like equality constraints at first order, whereas inactive inequalities should re-enter the feasible domain and remain nonbinding (Zhang et al., 2018).

Pathwise versus law constraints: Pathwise conditions restrict almost every realization, as in $g_I(x(t_f),t_f)\le 0$01 or $g_I(x(t_f),t_f)\le 0$02. Law constraints only restrict $g_I(x(t_f),t_f)\le 0$03, as in $g_I(x(t_f),t_f)\le 0$04, and therefore induce adjoint terminal conditions at the level of measure derivatives (Daudin, 2020). This suggests a separation between endpoint feasibility of trajectories and endpoint feasibility of distributions.

Terminal sets versus terminal equalities in MPC: Some MPC formulations use a terminal invariant set, admissible region, or safe set, such as $g_I(x(t_f),t_f)\le 0$05, rather than a pointwise equality (Dokania et al., 7 May 2026). Others enforce a tolerance box in error space or a learned safe set approximation (Wang et al., 21 May 2026). A plausible implication is that, in constrained receding-horizon control, terminal constraints often function more as recoverability certificates than as precise target equations.

Feasibility restoration versus feasibility preservation: Feasible-domain VEM preserves terminal feasibility once attained (Zhang et al., 2018). MEPDE restores it from infeasible initial guesses (Zhang et al., 2018). Terminal constraint handling thus depends strongly on whether the algorithm assumes feasible initialization.

Terminal constraints as structural sources of singularity: In stochastic LQ control, exact terminal subspace constraints, strict liquidation conditions, and partial infinite penalties all generate singular terminal behavior in Riccati equations, BSDEs, or decoupling fields (Ackermann et al., 7 Jan 2026). This suggests that singular terminal conditions are not technical artifacts but analytic encodings of hard endpoint restrictions.

Terminal constraints therefore constitute a unifying endpoint mechanism across multiple control theories, but not a single mathematical object. Depending on the setting, they may be terminal equalities, terminal inequalities, admissible terminal regions, invariant recovery sets, law constraints, random-subspace constraints, or penalized surrogates. The cited literature consistently treats them as decisive structural elements rather than optional boundary decorations: they determine optimality systems, feasibility logic, reachable sets, solver architecture, and, in stochastic settings, the singular terminal behavior of the adjoint equations.