Optimal trajectory generation for next generation flight management systems

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Starting from the 1st order necessary conditions for optimality, we derive a Hamiltonian system for the state and adjoints with split boundary conditions. Then, recognizing the two point boundary value problem as a canonical transformation, we employ generating functions to find the optimal feedback control as well as the optimal trajectory. Though we formulate the optimal control problem in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a non-linear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two point boundary value problem using a forward shooting method.

Difference inclusions provide a discrete-time analogue of differential inclusions, which in turn play an important role in the theories of optimal control, implicit differential equations, and invariance and viability, to name a few. In this paper we: (i) introduce a framework suitable for the study of difference inclusions for which the state evolves on a manifold; (ii) use this framework to develop necessary conditions for optimality for a broad class of discrete-time problems of dynamic optimization in which the state evolves on a manifold M. The necessary conditions for optimality we derive include the case for which the state $$q_i$$ is subject to constraints $$q_i \in S_i \subseteq M$$ , for $$S_i$$ a closed set. The resulting necessary conditions for optimality appear as discrete-time versions of the Euler–Lagrange inclusion studied by Ioffe (in Trans Am Math Soc 349(7):2871–2900, 1997), Ioffe and Rockafellar (in Calc Var Partial Differ Equ 4(1):59–87, 1996), Mordukhovich (in SIAM J Control Optim 33(3):882–915, 1995), Mordukhovich (in Variational analysis and generalized differentiation II: applications. Springer, Berlin, 2006), and Vinter and Zheng (in SIAM J Control Optim 35(1):56–77, 1997) generalized in a natural way to the case in which the state is evolving on a manifold.

We designed and tested a Special Aircraft and Air-crew Authorization Required (SAAAR), also known as Required Navigation Performance (RNP) Authorization Required (AR), to Localizer Performance with Vertical guidance (LPV) supported by the space-based augmentation system of GPS approach for runway 33 of Salzburg Airport, Austria. Approach to landing on runway 33 is severely impacted by mountainous terrain to the south of the airport that lies within the runway extended centerline, rendering all straight-in approaches but those based on RNP AR impossible. That is due to the high navigation accuracy available under RNP AR outside of the final approach segment. It makes it possible to keep the obstacle protection areas, which are constructed around the approach path and must be clear of obstacles to ensure the required obstacle clearance, very small so that they do not extend into the terrain. The combination of RNP AR and LPV, in turn, makes it possible to use the more precise angular guidance for the final approach available under LPV while still being able to exploit the significantly smaller RNP AR protection areas in all approach segments. In Salzburg, the resulting improved performance enables us to reduce the minima from 368 ft to 218 ft above runway threshold level. For the intermediate approach, we use curved legs based on a constant radius between two fixes (Radius to Fix legs) that lead onto the final approach course. The approach is coded as advanced RNP with 0.1 nautical mile values for the 95-percent lateral accuracy (RNP 0.1) on each sequence. Level D full flight simulator tests with an Airbus A350 showed that this coding is sufficient to achieve RNP 0.1 performance under all permitted conditions with the approach being presented to the pilots as RNP AR.

Necessary conditions for optimality in problems with nonstandard cost functionals.- Linear control problems and an existence theorem.- Optimal disturbance rejection and performance robustness in linear systems.- Necessary conditions for optimal disturbance rejection in linear systems.- Synthesis of finite-interval H ? controllers by state space methods.- Worst-case performance measures for linear control problems.- Model reduction with a finite-interval H ? criterion.

In this study, optimal control of a senior coefficient in a heat conduction problem is discussed. Since such problems are severely ill-posed, it is necessary to regularize them. So, we present the regularization process which has a great importance for a optimal control problem in this study. After proving the existence and uniqueness of optimal solution, a necessary condition for optimality is stated. Then, an iterative algorithm is purposed for numerical approximations. Finally, optimal control has been approximately calculated using the obtained results. The outcomes have also been tested with numerical examples.

This paper concerns optimal control of systems governed by stochastic partial differential equations in which drift and diffusion terms are second- and first-order differential operators, respectively. Necessary conditions for an optimal control are derived for both nondegenerate and degenerate systems, and all the coefficients appearing in the equations are allowed to depend on the control variables. Furthermore, the results obtained in the paper can also be used to derive necessary conditions of optimality for partially observed diffusions with correlation between the signals and the observation noises.

The present paper is dedicated to the investigation of optimal control problems whose behavior is described by quasilinear differential equations of first order on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem of the existence and uniqueness of a generalized solution in the space \(C_{\mu }(\overline{G})\) is proved for quasilinear differential equations; necessary conditions of optimality are obtained in terms of the principle of maximum; the Bitsadze–Samarski boundary value problem is proved for a linear differential equation of first order; the existence of a solution in the space \(C_{\mu }^{p}(\overline{G})\) is proved and an a priori estimate is derived. A theorem on the necessary and sufficient condition of optimality is proved for a linear optimal control problem.

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.

In this paper, the optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.

In this work, we discuss normal forms of necessary conditions of optimality (NCO) for optimal control problems subject to pathwise state constraints and for problems in the calculus of variations with inequality constraints. It is known that standard forms of the NCO may fail to provide information that is useful to identify optimal solutions, namely when the multiplier associated with the objective function takes the value zero. The normal forms of the NCO guarantee that the conditions remain always informative, which is of importance in critical applications where decisions based on optimization are taken, such as autonomous systems. Based on a previous nondegenerate maximum principle for optimal control problems, we extend the strengthness of these conditions to normality while applying them to the particular case of calculus of variations problems. We compare our results with existent normal forms of NCO for dynamic optimization problems and conclude that, when applied to calculus of variations problems, we may say that, under similar conditions, we can apply such result to a wider class of problems, having less regularity on the data.

The conventional dynamic programming methodology for the solution of optimal control, despite having many desirable features, is severely restricted by its computational requirements. However, in recent times, an alternate formulation, known as the adaptive-critic synthesis, has given it a new perspective. In this paper, we have attempted to use the philosophy of adaptive-critic design to the optimal control of distributed parameter systems. An important contribution of this study is the derivation of the necessary conditions of optimality for distributed parameter systems, described in discrete domain, following the principle of approximate dynamic programming. Then the derived necessary conditions of optimality are used to synthesize infinite time optimal neuro-controllers in the framework of adaptive-critic design. A motivating example that follows clearly shows the potential of the adaptive critic procedure.

Recently Zolezzi, [4] presented a necessary condition for optimality for the problem of optimal control of systems governed by parabolic partial differential equations with a first boundary condition. In this paper, we have developed (Theorem 4.3 and Corollary 4.5) a similar necessary condition for Cauchy problems (unbounded domains) that arise naturally in the optimal control problems of stochastic Ito differential systems with fixed terminal time. For the proof of our results we use several theorems of Aronson [1] which have been slightly modified to suit our requirements.

An extension of the classical Pontryagin maximum principle for Mayer problems without terminal constraints, subject to affine control systems $$\dot x = X_0 (x) + \sum\nolimits_{j = 1}^m u _j X_j (x)$$ , is proved. In connection with a suitable dilation on the state space ℝ n , we introduce a class of “homogeneous tangent vectors” which provide a nonlinear, high-order approximation of the attainable set in the case where the usual linear approximation proves to be inadequate. By studying control variations which generate homogeneous tangent vectors, we derive new necessary conditions for optimality that are particularly effective for basically nonlinear optimal control problems where other high-order tests provide no conclusive information.

The derivation of necessary conditions of optimality for optimal control problems with state/control constraints is a challenging task, not always clearly understood. In this paper, we present a didactic approach based on the method of exponential penalty functions allowing, in our setting, to significantly simplify the proofs. This work may be viewed as a proposal of a course designed for students familiar with the basics of optimal control theory and elements of functional analysis. To make the analysis as simple as possible, we concentrate on the case of optimal control problem with fixed initial point and free end state.

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.