Structure of optimal control for planetary landing with control and state constraints

Abstract

This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc.

Optimal planetary landing with pointing and glide-slope constraints

“…We deduce that d z can take only two values, cos(θ) and the constant value |δ z | (if this value belongs to (cos(θ), 1]), and can change value at most one time. This contradicts (12). Thus p 0 and q r are not collinear.…”

“…The complete proof of Theorem 1 is too long and technical to be presented in this paper ; it is detailed in [12]. Here we report the main steps we went through to achieve it.…”

“…This implies that ⟨q r , d⟩ is not differentiable when the state constraint is active. Consequently, to go to the conclusion, it is necessary to show in the first hand that the derivative of ⟨q r , d⟩ is negative where it exists, and on the other hand that the discontinuities of ⟨q r , d⟩ do not cause additional sign changes (see [12]).…”

“…They seldom arise when solving landing problems numerically, and the next lemma states that they do not exist in the case of an altitude constraint (γ = 0) if the initial conditions are generic, which is defined in Lemma 3. The result of Lemma 3 has been proved in [12] under the two following assumptions.…”

“…In this framework, we give elements to show that the optimal control has either a Max-Min-Max form or a Max-Singular-Max form. The complete proof on the structure of the optimal control is too long to be detailed here, but it is presented in [12]. Moreover, we analyze a particular case, in which the mass is assumed to be constant and the glide-slope constraint reduced to a positive altitude constraint.…”