Linear Stochastic Optimization via Differential Algebra

• Linear Stochastic Optimization with Differential Algebra is a method that uses differential algebra to linearize stochastic differential equations and propagate uncertainties in optimal control problems.
• It employs automatic differentiation to extract Jacobians, propagate state covariances, and transcribe chance constraints using a Gaussian approximation.
• Applications include trajectory optimization in space missions, robotics, and stochastic control, demonstrating robust performance under moderate uncertainties.

Linear Stochastic Optimization with Differential Algebra (L-SODA) encompasses a class of methods and theoretical results leveraging differential algebraic techniques to solve optimization problems constrained by linear stochastic differential equations, often with uncertainty propagation, chance constraints, and algebraic structure. It constitutes an efficient regime for control, estimation, and trajectory design where state and control uncertainties are sufficiently small to justify linearization of the underlying stochastic dynamics and constraints, typically facilitated with automatic differentiation and high-order polynomial expansions via differential algebra.

1. Problem Formulation and Objectives

L-SODA addresses optimal control and trajectory optimization of systems governed by linear stochastic dynamics, enforcing chance constraints in a probabilistic framework. The central optimization problem can be abstracted as:

$$\begin{align*} & \text{minimize} \quad J(x_0, u_{0:T-1}) \ & \text{subject to} \quad x_{k+1} = f_{k}(x_k, u_k), \quad \forall k\ & \mathbb{P}\left( h(x_k, u_k) \preceq 0 \right) \geq 1 - \beta \quad \forall k \end{align*}$$

where $x_k, u_k$ are state and control random variables, $f_k$ is linear or locally linearized via DA, and $h(\cdot)$ encodes path or terminal constraints transcribed in chance-constrained fashion with parameter $\beta$.

Differential algebra is employed to generate high-order Taylor expansions of $f_k$ and $h(\cdot)$, but within L-SODA, only first-order (linear) terms are used to propagate uncertainty, yielding a Gaussian approximation for the state distribution.

2. Mathematical Structure and Differential Algebra Framework

The propagation of moments utilizes linearization:

• The state transition is expressed as:

$x_{k+1} = A_k x_k + B_k u_k + w_k,$

where $A_k$ and $B_k$ are Jacobian matrices extracted efficiently via DA; $w_k$ encodes process noise.

• Covariances are updated by:

$$\Sigma_{x, k+1} = A_k \Sigma_{x, k} A_k^\top + Q_k,$$

where $Q_k$ is the process noise covariance.

• Chance constraints on $y = h(x_k, u_k)$ are encoded by demanding:

$$y \sim \mathcal{N}(\bar{y}, \Sigma_y), \quad \bar{y} = h(\bar{x}_k, \bar{u}_k), \quad \Sigma_y = J_h \Sigma_{x, k} J_h^\top,$$

with $J_h$ the Jacobian. The probabilistic constraint $\mathbb{P}(y \preceq 0) \geq 1 - \beta$ is transcribed as:

$\bar{y} + \Psi_d^{-1}(\beta) \sigma_y \preceq 0,$

using the suitable quantile transform ($\Psi_d^{-1}(\beta)$), and $\sigma_y$ the diagonal square roots of $\Sigma_y$.

In the context of DA, the first-order expansion is extracted as: $$h(x_k, u_k) \approx h(\bar{x}_k, \bar{u}_k) + J_h (x_k - \bar{x}_k),$$ where higher-order terms are omitted due to small uncertainty assumptions.

3. Core Algorithms and Implementation Aspects

The deterministic transcription of chance constraints allows for efficient solution by standard convex optimization or optimal control methods. A typical L-SODA workflow is:

1. Nominal trajectory optimization: Solve for $(\bar{x}_k, \bar{u}_k)$ neglecting uncertainty.
2. Uncertainty propagation: At each time step, propagate $\Sigma_{x, k}$ linearly using DA-extracted Jacobians.
3. Constraint evaluation: Apply first-order chance constraint transcription, tightening constraints by safety margins proportional to $\Psi_d^{-1}(\beta)$ and $\sigma_y$.
4. Iterative refinement: Update nominal and uncertainty trajectories as needed.

For example, in the stochastic double integrator benchmark, L-SODA was shown to converge in a single iteration, with state and control uncertainties finely accounted for, and with final constraints satisfied at the prescribed risk level.

4. Theoretical Guarantees and Comparison with Nonlinear Frameworks

The key theoretical justification for L-SODA's approach derives from the linearity of uncertainty propagation under small deviation regimes. The DA framework ensures rigorous, automatic extraction of derivatives requisite for covariance propagation, and the Gaussian assumption remains robust due to limited nonlinearity.

In contrast, the full nonlinear SODA solver allocates risk across adaptively decomposed Gaussian mixture components, propagating non-Gaussian uncertainties via domain splitting (LOADS), and transcribes constraints component-wise—resulting in increased computational burden and robustness for large uncertainties or strongly nonlinear dynamics.

The linear regime's deterministic performance recovery, validated on fuel-optimal Earth-Mars transfer scenarios and similar problems, demonstrates nearly identical results versus the classic deterministic optimization, with only marginal additional resource consumption.

5. Handling of Stochastic Differential Equations and Control

L-SODA is closely connected to advances in the theory of linear stochastic differential equations and control under uncertainty. It benefits from the existence and uniqueness theory extended for linear SDEs driven by Lévy processes (León et al., 2012), where the anticipative Girsanov transformation and Malliavin calculus allow closed-form representations for stochastic processes even when coefficients and initial conditions are non-adapted and random.

These theoretical results permit handling of systems with jumps, random, and potentially anticipative coefficients, facilitating optimization in more general settings where not only Gaussian noise but Poisson-driven events may influence the solution.

In stochastic control applications (see also (Wang et al., 2016)), LP-exact controllability and equivalence with observability and functional optimization problems support synthesis of controllers that direct the system exactly to target distributions under prescribed risk thresholds—directly applicable in L-SODA's chance-constrained context.

6. Applications and Validation

L-SODA provides substantial computational advantages in:

• Preliminary trajectory design for space missions (low-thrust, interplanetary transfers, rendezvous under navigation error).
• Robotics and autonomous vehicles, where high accuracy and online compliance with probabilistic safety constraints are needed.
• Stochastic control for linear systems with algebraic or differential-algebraic constraints, especially when solving for the optimal control laws under uncertainty.

In benchmark cases, such as stochastic double integrator and Earth–Mars transfer, Monte Carlo validation revealed that L-SODA achieves constraint satisfaction (e.g., failure risk of a few percent) with minimal excess resource usage, and matches deterministic performance tightly.

7. Research Directions and Limitations

L-SODA is most accurate when uncertainties are tightly bounded and system nonlinearities are negligible in the local domain of interest. Its main limitations stem from:

• Reduced accuracy in strongly nonlinear regimes or with non-Gaussian, heavy-tailed uncertainty.
• Necessity to verify the validity of the linear approximation via DA—a region outside which SODA or related nonlinear tools should be invoked.

Future research avenues include extension to nonlinear SDEs with jumps and anticipation (León et al., 2012), robust design against coefficient perturbations, and hybrid or hierarchical solvers that switch between linear and nonlinear uncertainty handling dynamically.

In summary, L-SODA constitutes a rigorous, DA-based, computationally efficient toolkit for chance-constrained optimization under linear stochastic dynamics, leveraging modern advances in stochastic analysis, automatic differentiation, and convex transcription to deliver robust solutions where uncertainty remains moderate and tractable.