arXiv subjects

Vitaly Shaferman

Publications and source records attributed to Vitaly Shaferman.

5 recordsLinked to original sources

arXiv2026

Low-Thrust Orbital Differential Games with Speed Constraint Enforcement Using Cost Weighting

This paper considers the problem of a low-thrust spacecraft pursuit-evasion differential game with an arbitrary terminal relative speed constraint. It addresses the terminal phase of the engagement for two relatively close spacecraft near a circular orbit. The problem is formulated as a linear-quadratic zero-sum differential game, with soft constraints on the terminal relative position and velocity, and running costs on the players' control efforts. An analytical, closed-loop, minimum-fuel-consumption optimal guidance law is derived for each player, forming a saddle-point solution. It is proven that any terminal speed can be achieved by properly choosing the weighting parameters of the cost function. To verify the optimality of the solution, a conjugate point analysis is performed when the cost function velocity weighting matrix is either positive or negative definite. The negative-definite case arises at high terminal speeds and is seldom seen in the literature. The performance of the derived guidance law is evaluated in simulations for different target maneuvers and compared to a state-of-the-art optimal-control-based guidance law. The simulations show that the derived guidance law satisfies the constraints and offers a substantial advantage over the optimal-control-based guidance law when the target is optimally evading.

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arXiv2026

Optimal Bounded Thrust Powered Descent with Analytical Ground-Collision Avoidance

The paper proposes a new approach to address the bounded-thrust powered-descent problem while ensuring ground-collision avoidance. A time-dependent polynomial approximation of the mass is employed to formulate a bounded linear-quadratic optimal-control problem that minimizes the thrust-acceleration control effort, terminal miss, and terminal velocity error. The resulting approximation is used to impose a hard constraint on the horizontal thrust profile while keeping the vertical thrust profile unconstrained. The key idea is a hierarchical separation of the thrust allocation, which enables analytical ground-collision avoidance under bounded thrust. Unlike existing bounded-thrust powered-descent approaches based on numerical optimization and trajectory-shaping constraints, the proposed method provides explicit analytical collision-avoidance conditions. Building on this formulation, the guidance law predicts the switching times between saturated and unsaturated arcs and shapes the thrust-acceleration profile to achieve a soft landing, even when the controller remains saturated over extended portions of the trajectory. Owing to its analytical nature, the guidance law is computationally efficient, and its continuous thrust profile facilitates real-time implementation. The proposed method was evaluated over a grid of perturbed initial conditions in realistic simulations, demonstrating accurate collision-free soft-landing performance. The results highlight the importance of combining saturation-aware guidance with ground-collision avoidance under bounded thrust.

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arXiv2026

Optimal Powered Descent Guidance with Pyramid-Shaped Approach-Angle Constraints

In this paper, a novel optimal soft-landing guidance law with inequality approach-angle path constraints is analytically derived. The proposed guidance law prevents ground collision and enables approach-angle control by constraining the optimal trajectory to remain within a convex inverted pyramid originating at the landing point. A 3D point-mass linear kinematic model in a constant gravitational field is employed, together with a quadratic control-effort cost and terminal constraints on position and velocity. Analytical open-loop and closed-loop solutions, together with the optimal final time, are derived using Pontryagin's Minimum Principle and the optimality conditions at the transitions between unconstrained and constrained arcs. It is additionally shown that the optimal final time decreases when the path constraints become active. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, the controller cancels the gravitational component normal to the constraint, causing the trajectory to evolve along the constraint surface. The proposed guidance law is evaluated in simulations under various initial conditions, demonstrating accurate landing performance and consistent satisfaction of the path constraints.

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arXiv2026

Optimal Ground-to-Air Interception with Time-Varying Acceleration Bounds

This paper proposes novel optimal-control-based guidance laws for ground-to-air missiles with time-varying acceleration bounds. In such engagements, as the missile climbs in altitude, its acceleration bound decreases, which may lead to acceleration saturation and significant miss distances if not explicitly accounted for. The proposed guidance laws incorporate hard acceleration command constraints directly into a linear-quadratic optimal-control framework, in contrast to conventional unbounded or softly constrained approaches. Analytically based guidance laws are developed for linear zero-order and first-order strictly proper missile dynamics with arbitrary-order linear target dynamics. Unlike the constant hard-bound case with minimum-phase missile dynamics, time-varying acceleration command bounds permit an initial unsaturated interval in which the proposed guidance laws can anticipate future saturation and reshape the acceleration profile accordingly. This enables earlier maneuvers when the missile possesses greater low-altitude maneuverability, fundamentally altering the structure of the optimal solution. The proposed approach is evaluated in nonlinear simulations and compared with equivalent unbounded and softly constrained optimal guidance laws. The results demonstrate substantially improved interception performance under saturation, reduced tuning requirements compared to softly constrained guidance laws, and enhanced capability in challenging engagement scenarios.

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arXiv2026

Flyby Distance Pursuit for Guarding a Target with an Inferior Guard

Guarding a target against a fast Attacker with a slower Guard is posed as a differential game. Both players follow nonlinear, thrust-free, aerodynamic motion in the plane. The capturability problem inherent in this scenario is addressed by formulating the game such that the Attacker is constrained to terminate at its Target, and the game value is the flyby distance. The latter is made accessible at termination by introducing an auxiliary state variable that records the players' minimum separation distance throughout their flight. This novel combination of termination condition and auxiliary state removes a structural limitation of current pursuit games and opens a new class of solvable problems. It also enables us to embed attacker mission constraints directly. Taking an indirect approach, we formulate the game's Two-Point Boundary Value Problem, solve it numerically, and identify several solution types. When the Attacker's only constraint is to reach the Target, he can do so without being captured. These types of solutions, however, are less practical due to long flight times and slow terminal speeds. Imposing an Attacker terminal speed constraint yields simpler solutions and reveals the trade-off between terminal speed and flyby distance. We demonstrate that in such a setting, capture is possible in a relevant region of the parameter space.

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