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2023 | OriginalPaper | Buchkapitel

Second-order Sufficient Conditions of Strong Minimality with applications to Orbital Transfers

verfasst von : Leonardo Mazzini

Erschienen in: Modeling and Optimization in Space Engineering

Verlag: Springer International Publishing

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Abstract

We present second-order Extended Sufficiency Conditions applicable without Strict Legendre Conditions and without local controllability assumption also in the frame of saturated and bang–bang control. We provide a procedure to maximize the interval where the Sufficient Conditions can be applied using a Riccati Matrix Equation and compare this procedure with the classic conjugate point condition. Applications to Finite and Infinite Thrust Orbital Transfer cases are given at the end.

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Vorheriges Kapitel Data Reduction for Optimizing the Attitude Control Dispatch in a Spacecraft

Nächstes Kapitel Evolutionary Optimisation of a Flexible-Launcher Simple Adaptive Control System

The symbol | | will be used to mean the Euclidean Norm.

Both hypotheses can be relaxed, and we introduce them in this form for simplicity.

We refer to the Clarke generalized gradient, see [10], which is a non-empty closed convex set; we use the same symbol to mean the generalized gradient set and the partial derivative, the ambiguity being resolved by the logical statement.

This representation structurally satisfies the Maximum Principle and is a general presentation of the H0/I Hamiltonians close to cross points. It is easy to verify that H(z) = H₀(z) − k(S)S(z) does not agree with the Maximum Principle.

To clarify the notation used, ∂_σ,σg = Σ^T∂_x,xg Σ. x_σ = Σ^Tx, so σ, π, ν, and μ are multi-indices that indicate the components of a matrix after a transformation by Σ, Π, N, M; 0_1,π = 0_ν,π = 0_1,dim(π), and I_σ,σ is the identity matrix of dimension dim(σ).

\(\partial _{\alpha ,\beta }\hat {H}_t\) are the second derivatives of the Hamiltonian on the candidate extremal arc.

Δ gives the jump of the variable in the time positive direction, i.e., \( \bar {x}|{ }_{t}=\bar {x}|{ }_{t+}-\bar {x}|{ }_{t-}\)

Calling as before in Eqs. (7) \(A_t=\partial _{xp}\hat {H}_t, B_t=\partial _{pp}\hat {H}_t,C_t=\partial _{xx}\hat {H}\).

A_t, B_t, and C_t are defined by Eqs. (7).

This is an extension of the classic ones which do not take into account the jumps at the cross points.

The solution at a given time t ∈ [t₀, t_f] − Θ can be seen as a finite chain of continuous functions. From the terminal point to the first crossing point, \((U^{\beta }_{t}, V^{\beta }_{t})\) are locally continuous with respect to β due to the classic ODE theorems and the continuity of A_t, B_t, and C_t, at the crossing point the left limit of the solution is continuous with respect to the right limit (see Eqs. (22)), and then we sequentially continue this process in a finite number of steps until we reach the point t.

The additional Normality condition requested in Ref.[12] is the Strong Normality of both terminals. The condition for the right terminal is obviously satisfied because Π = ⊘. The condition for the left terminal can be easily proved because \(p^T f(\hat {x},\hat {u})- p^T f(\hat {x},u)= p_3(\hat {p}_3/\chi -u_s)+p_4(\hat {p}_4/\chi -u_c) \geq 0, \forall (u_s,u_c) \in U\) admits as only solution p = 0 (it suffices to choose u_s = 2p₃∕χ, u_c = 2p₄∕χ to make this expression strictly negative for any p ≠ 0).

[1]

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[2]

V. Zeidan, Extended Jacobi Sufficiency Criterion for Optimal Control, SIAM, Journal of Control and Optimization, Vol.22, No 2, Mar. 1984.

[3]

V. Zeidan, Sufficiency Conditions with Minimal Regularity Assumptions, Applied Mathematics and Optimization 20:19–31 (1989), Springer Verlag New York.

[4]

V. Zeidan, Sufficient Conditions for the generalized problem of Bolza, Transactions of the American Mathematical Society, Vol.275, Number 2, February 1983.

[5]

D.G. Hull, Sufficient Conditions for a Minimum of the Free-Final-Time Optimal Control Problem, JOTA, Vol. 68, No. 2, Feb. 1991.

[6]

D.G. Hull, Optimal Control Theory for Applications, 2003 Springer-Verlag New York.

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[8]

H. Schattler, U. Ledzewicz , Geometric Optimal Control: Theory, Methods and Examples, Interdisciplinary Applied Mathematics 38, Springer (2012)

[9]

A.A.Agrachev, G. Stefani, P.Zezza , Strong Optimality for a Bang-Bang Trajectory, SIAM J. Control and Optimization (2002),Vol. 41, No. 4, pp. 991–1014MathSciNetCrossRefMATH

[10]

F.H. Clarke, Generalized Gradients and Applications, Transactions of the America Mathematical Society, Vol. 2015,1975

[11]

N.P.Osmolovskji, H.Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM, 2012CrossRef

[12]

V. Zeidan, P. Zezza , The Conjugate point condition for smooth control sets, Journal of Mathematical Analysis ans Applications, 572–589 (1988)

[13]

A.F. Filippov, Differential Equations with Discontinuous Right Hand Sides, Springer, 1988CrossRefMATH

[14]

J.W. Jo, J. E. Prussing, Procedure for Applying Second-Order Conditions in Optimal Control Problems, Journal of Guidance, Control and Dynamics, Vol. 23, No.2, March-April 2000

[15]

L.J. Wood, Sufficient conditions for a local minimum of the Bolza Problem with multiple terminal point constraints, AAS/AIAA Astrodynamics Specialists Conference, Durango, Colorado August 19–22, 1991, AAS Publication Office, P.O. Box 28130, San Diego CA 92128

[16]

L.Mazzini, Time Open Orbital Transfer in a Transformed Hamiltonian Setting, Journal of Guidance, Control and Dynamics, Vol. 36, No. 5, September October 2013

[17]

L. Mazzini, Finite Thrust Orbital Transfers, Acta Astronautica, AA-D-13-00699, 2014

[18]

L. Mazzini, Flexible Spacecraft Dynamics, Control and Guidance, Springer, 2016CrossRefMATH

Titel: Second-order Sufficient Conditions of Strong Minimality with applications to Orbital Transfers
verfasst von: Leonardo Mazzini
Verlag: Springer International Publishing
Buch: Modeling and Optimization in Space Engineering
Print ISBN: 978-3-031-24811-5

Electronic ISBN: 978-3-031-24812-2

Copyright-Jahr: 2023
DOI: https://doi.org/10.1007/978-3-031-24812-2_8

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