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Neighboring Optimal Guidance in Bang Bang Control with Target

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Abstract

The purpose of this paper is to present a Neighboring Optimal Guidance algorithm capable of driving a dynamical system along an optimal trajectory to a target when the Hamiltonian fails to satisfy the strict Legendre condition. Similar problems are associated with saturated and bang–bang controls as, for instance, in the case of low thrust orbit transfer problems. The approach is based on a new formulation of the second-order sufficient conditions for optimality—introduced by the author—which make tractable, problems with irregular Hamiltonians. Effectiveness of the guidance scheme proposed in this work is successfully tested on a space mission scenario, i.e. Neighboring Optimal Guidance in a low-thrust orbit transfer about Earth.

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Notes

  1. The Legendre strict conditions, request that the pre-Hamiltonian second derivative in the controls is strictly negative, that is \(\partial _{uu}H_P <0\) in the notation which will follow.

  2. AC is the set of Absolutely Continuous functions, \(L^{\infty }\) is the set of Essentially Limited functions.

  3. The symbol \(| \cdot |\) is used to mean the Euclidean Norm.

  4. The Legendre–Clebsch conditions in terms of the pre-Hamiltonian are \(\partial _{uu}H_P < 0\)

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

  6. We refer to the Clarke generalized gradient see [6], 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.

  7. 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 \(H(z)=H_0(z)-k(S) S(z)\) does not agree with the Maximum Principle.

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

  9. A field of extremals is a set of characteristics parameterized by \(\xi \in {\mathbb {R}}^{n}\) such that \( \oint P \partial _{\xi } X d\xi =0\) in any closed loop of \({\mathbb {R}}^n\). It allows to propagate a solution of the Hamilton Jacobi Bellman Equation as described in Ref. [16] Proposition 5.1. Such solutions can be obtained by the final conditions using the characteristics \(\phi (X(t,\xi ),t)= \phi (X(t_f,\xi ),t_f),\partial _X \phi (X,t)= P(t,\xi )\).

  10. The classic conditions of \(\varPi \) controllability when the Legendre–Clebsh conditions are satisfied are (see Ref. [27, Lemma 5.3.3]): there is no \(p_t\) such that \(\dot{p}_t=-\partial _x f^T p\), with \(p_{t_f}=\) range\((\varPi )\) and \(p^T_t \partial _u f=0\) in \([t,t_f]\). When this condition is not verified it follows easily that \(U^{\eta }_t \varPi \) is defective and \((\varPi ^T U^{{\mathcal {N}}-1}_{t} U^{\eta }_t \varPi )\) is not invertible. This can be seen using the general expressions of \(A=\partial _x f + \partial _u f \partial _x u\), \(A^T=\partial _x f^T+ \partial _x u^T \partial _u f^T\) and \(B=B^T=\partial _u f \partial _p u =\partial _p u^T \partial _u f^T\). Where \(\partial _p u\) and \(\partial _x u\) are derived from the feedback law. If the classic conditions are verified then there is a costate \(p^* \in \) range\((\varPi )\) so that \(U^{\eta }_t p^*=0\).

  11. The capability to measure the state is assumed in the NOG algorithm.

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  1. Thales Alenia Space, Rome, Italy

    Leonardo Mazzini

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  1. Leonardo Mazzini
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Correspondence to Leonardo Mazzini.

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Communicated by Ryan Russell.

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Mazzini, L. Neighboring Optimal Guidance in Bang Bang Control with Target. J Optim Theory Appl 199, 310–336 (2023). https://doi.org/10.1007/s10957-023-02286-1

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