Monotone Cost Shaping in Optimal Control

• Monotone Cost Shaping is a method that embeds time-varying, monotonic weight profiles into stage costs to ensure closed-loop stability and performance in control systems.
• It adapts the stage cost in MPC and LQR by replacing terminal constraints with polynomial or quadratic shaping terms, yielding quantifiable improvements in stability.
• MCS facilitates robust performance across applications, including nonlinear MPC and domain-specific designs like energy-optimal AUV navigation in dynamic environments.

Monotone Cost Shaping (MCS) refers to a family of methods for modifying the stage cost in optimal control and @@@@1@@@@ (MPC) by embedding a monotonic, typically time-varying, weight profile or cost-shaping term within the objective. The core principle is to induce or exploit properties such as closed-loop stability, monotonic improvement, or task-driven behavioral bias without relying on explicit terminal constraints or structural modifications to the system dynamics. MCS has been developed and analyzed for nonlinear finite-horizon MPC, infinite-horizon LQR, nonlinear control-affine systems, and application-specific domains such as energy-optimal autonomous underwater vehicle (AUV) navigation in dynamic environments.

1. Foundational Principles of Monotone Cost Shaping

Monotone Cost Shaping modifies the finite-horizon or infinite-horizon optimal control criterion by introducing a monotonically increasing or nonnegative shaping term to the stage-wise cost. In the canonical nonlinear MPC setting, the approach is to replace explicit terminal-state constraints and terminal costs with a time-varying polynomial weight profile on the stage cost. For a controlled system with state $x_k \in \mathbb{R}^n$ and input $u_k \in \mathbb{R}^{n_u}$ evolving as $x_{k+1} = f(x_k,u_k)$, and a positive-definite stage cost $\ell(x)$, the MCS-MPC cost is

$$J_m(u|x_k) = \sum_{i=1}^N w_i\,\ell(x^i_u(x_k)),\qquad w_i = (i/N)^m,\,\quad i=1,\ldots,N$$

where $m\in\mathbb{N}$ controls the growth rate of the weighting profile from $w_1$ to $w_N=1$. By making $w_i$ sharply increasing near the horizon, the tail of the trajectory is penalized heavily, providing a Lyapunov-like decrease in the cost and asymptotic stability without a terminal constraint (Alamir, 2017).

In the linear quadratic regulator (LQR) context, Monotone Cost Shaping adds a parameterized, positive semi-definite quadratic form to the stage cost:

$J_\alpha = J_0 + \alpha\,\sum_{t=0}^{\infty} \gamma^t\,\phi(x_t, u_t),\qquad \phi(x,u) = \frac{1}{2} \begin{pmatrix}x\u\end{pmatrix}^T H \begin{pmatrix}x\u\end{pmatrix}$

where $H \succeq 0$ and $\alpha\geq 0$. The shaped solution has optimal gains and value determined by modified Riccati and feedback expressions, guaranteeing monotonic improvement in the closed-loop cost under spectral and definiteness conditions (Rai et al., 11 Oct 2025).

2. Stability and Performance Guarantees

The most fundamental property induced by MCS in nonlinear predictive control is asymptotic stability of the system origin on a predesignated compact set $\mathbb{X}_N$. Provided the stage cost $\ell$ is continuous and positive-definite, and the zero state is $N$-step reachable, MCS allows the closed loop

$x_{k+1} = f\left(x_k,\,u^*_0(x_k, m)\right)$

to be stabilized by sufficiently large $m$. The optimal cost $V(x) = J_m^*(x)$ serves as a Lyapunov function satisfying

$$V(f(x, u^*_0)) - V(x) \leq -\frac{1}{N^m}\ell(x_1) - \frac{\gamma}{2}\ell(x_N^*(x))$$

yielding $V(x_{k+1}) \leq V(x_k) - \alpha(\|x_k\|)$ for some positive definite $\alpha$ (Alamir, 2017).

For infinite-horizon LQR with shaping, provided $H \succeq 0$, $Q \succeq 0$, $R \succ 0$, and the shaped Riccati equation is solvable, the closed-form feedback gain $K(\alpha)$ is monotone nondecreasing in $\alpha$ in the Loewner (semidefinite) order, and the resulting system is globally asymptotically stable. In nonlinear control-affine systems, cost shaping with an auxiliary term $v(x,\theta)$ and associated PDE for the value perturbation $h(x,\theta)$ guarantees global and input-to-state stability for the closed loop if the original controller is stable (Rai et al., 11 Oct 2025).

3. Weight Profile Design and Tuning Criteria

In finite-horizon MPC, the weight profile $w_i = (i/N)^m$ must satisfy strict monotonicity and growth conditions:

• Monotonicity: $w_{i+1} \geq w_i$ for $i = 1,\ldots,N-1$.
• Exponential growth: The ratio $w_{i+1} / w_i = ((i+1)/i)^m$ becomes large as $m$ increases.
• Growth factor: With $c = (N-1)/N$, set $\psi(m) = 1 - c^m$; as $m \to \infty$, $\psi(m) \to 1$.

Explicit lower bounds on $m$ guarantee that the state reaches the domain of the local control Lyapunov property, ensuring that the overall scheme is stabilizing:

$$m \geq \max\left\{ \frac{\ln(\bar\rho/\eta)}{\ln c},\,\frac{\ln(\gamma/2)}{\ln c} \right\}$$

where $\eta$ is the supremum over intermediate stage costs and $\bar\rho$, $\gamma$ come from the local Lyapunov decrease assumption.

Practical tuning thus proceeds by:

• Selecting the smallest $N$ with $N$-step reachability,
• Choosing $\ell(x)$ and local $q(x)\geq\gamma\ell(x)$,
• Estimating $\eta$,
• Computing $m$ as above,
• Implementing the cost profile (Alamir, 2017).

4. Application of MCS in Nonlinear and Linear-Quadratic Settings

The scope of Monotone Cost Shaping includes:

• Nonlinear finite-horizon MPC: MCS replaces terminal constraints/costs by polynomial weights; proven stability under standard regularity and reachability assumptions (Alamir, 2017).
• Infinite-horizon LQR: Additional positive semidefinite terms shape the optimal feedback law. The explicit formula for the shaped gain is

$$K(\alpha) = [R(\alpha) + \gamma B^T P(\alpha) B]^{-1} [\gamma B^T P(\alpha)A + M(\alpha)^T]$$

where $Q(\alpha),\,R(\alpha),\,M(\alpha)$ are augmented by the shaping term (Rai et al., 11 Oct 2025).

• Nonlinear control-affine systems: By enforcing the value perturbation PDE for the shaping term, the optimal feedback law shifts by $-\frac{1}{2}R^{-1}g^T\nabla h(x,\theta)$ and preserves stability and, under mild scaling, input-to-state stability.
• Closed-loop performance index shaping: A general framework for analytical linking and systematic shaping of the performance index, including an efficient gradient-based tuning algorithm for trajectory-level objectives, with monotone improvement guaranteed at each step (Rai et al., 11 Oct 2025).

5. MCS in Domain-Specific Predictive Control: Ocean Current-Aware MPC

In domain-adapted predictive control, MCS enables exploitation of task-relevant environment features. For energy-optimal AUV navigation in time-varying ocean currents, MCS is integrated into a stage-gated MPC objective where a real-time "helpfulness" scalar $s_k$ (measuring alignment and strength of current with the goal direction) gates relaxation and energy rebate terms:

$$\Delta\ell_k^{\text{MCS}} = -s_k\left[\lambda_{\text{relax}} \|e_{\parallel,k}\|_{Q_{\text{pos}}}^2 + w_{\text{reb}} \phi(u_{k,\mathrm{lin}})\right]$$

with $0 \leq s_k, \lambda_{\text{relax}} \leq 1$, and $\phi(u_{k,\mathrm{lin}})$ a bounded rebate function. This construction decreases the objective whenever $s_k>0$, ensuring that MCS is non-worsening,

$$J_{\text{base}} + \sum_k \Delta\ell_k^{\text{MCS}} \leq J_{\text{base}}$$

and provides explicit bounds on energy rebate per stage. The entire cost is $C^1$, tunable, and compatible with standard NLP-based MPC frameworks. Additional terms, such as Speed-to-Fly (STF) shaping, can be composed with MCS (Syntakas et al., 31 Jan 2026).

6. Theoretical Context and Analytical Properties

MCS methods preserve or enhance optimal control properties without the complexity of terminal constraint design. In LQR and affine-nonlinear settings, the induced feedback law's monotonicity with respect to shaping parameterization is analytically verifiable through explicit Riccati and Lyapunov analyses:

• All gain and value mappings are monotone in the shaping parameter under definiteness conditions.
• In the nonlinear case, the closed-loop system remains globally asymptotically stable and ISS, as proven via construction of an ISS-Lyapunov function and PDE-constrained value perturbation (Rai et al., 11 Oct 2025).
• For gradient-based tuning, each update step is guaranteed to monotonically improve the selected trajectory-level performance metric.

7. Summary Table: MCS Formulations

Context	Stage/Objective	Shaping Mechanism
Nonlinear MPC (Alamir, 2017)	$J_m(u|x)=\sum_{i=1}^N w_i \ell(x^i)$	Polynomial increasing weight, $w_i = (i/N)^m$
Infinite-Horizon LQR (Rai et al., 11 Oct 2025)	$$J_\alpha = J_0 + \alpha \!\!\sum \gamma^t \phi(x_t,u_t)$$	Parametric quadratic, $\phi$ with $H \succeq 0$
Nonlinear affine (Rai et al., 11 Oct 2025)	$V_0 + \int v(x,\theta)dt$	Gradient-based shaping via Lyapunov PDE
Ocean-aware MPC (Syntakas et al., 31 Jan 2026)	Baseline $+$ help-gated rebates	Stagewise gate $s_k$, relaxation, energy rebate

Each MCS variant maintains or improves closed-loop performance under established stability, monotonicity, and robustness criteria, with explicit tuning rules, non-worsening properties, and smooth parameterizations suitable for first-principles analysis and domain-specific design.