Optimal Interplanetary Orbital Transfers via Electrical Engines

The Hohmann transfer theory, developed in the 19th century, is the kernel of orbital transfer with minimum propellant mass by means of chemical engines. The success of the Deep Space 1 spacecraft has paved the way toward using advanced electrical engines in space. While chemical engines are characterized by high thrust and low specific impulse, electrical engines are characterized by low thrust and hight specific impulse. In this paper, we focus on four issues of optimal interplanetary transfer for a spacecraft powered by an electrical engine controlled via the thrust direction and thrust setting: (a) trajectories of compromise between transfer time and propellant mass, (b) trajectories of minimum time, (c) trajectories of minimum propellant mass, and (d) relations with the Hohmann transfer trajectory. The resulting fundamental properties are as follows:

Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method

In the last decades, heuristic techniques have become established as suitable approaches for solving optimal control problems. Unlike deterministic methods, they do not suffer from locality of the results and do not require any starting guess to yield an optimal solution. The main disadvantages of heuristic algorithms are the lack of any convergence proof and the capability of yielding only a near optimal solution, if a particular representation for control variables is adopted. This paper describes the indirect swarming method, based on the joint use of the analytical necessary conditions for optimality, together with a simple heuristic technique, namely the particle swarm algorithm. This methodology circumvents the previously mentioned disadvantages of using heuristic approaches, while retaining their advantageous feature of not requiring any starting guess to generate an optimal solution. The particle swarm algorithm is chosen among the different available heuristic techniques, due to its apparent simplicity and the recent promising results reported in the scientific literature. Two different orbital maneuvering problems are considered and solved with great numerical accuracy, and this testifies to the effectiveness of the indirect swarming algorithm in solving low-thrust trajectory optimization problems.     

Optimal Ascent Trajectories and Feasibility of Next-Generation Orbital Spacecraft

This paper deals with the optimization of the ascent trajectories for single-stage-to-orbit (SSTO) and two-stage-to-orbit (TSTO) rocket-powered spacecraft. The maximum payload weight problem is studied for various combinations of initial thrust-to-weight ratio, engine specific impulse, and spacecraft structural factor. For TSTO rocket-powered spacecraft, two cases are studied: uniform structural factor and nonuniform structural factor between stages.The main conclusions are that: the design of SSTO configurations might be comfortably feasible, marginally feasible, or unfeasible, depending on the parameter values assumed; the design of TSTO configurations is not only feasible, but the payload appears to be considerably larger than that of SSTO configurations; for the case of a nonuniform structural factor, the most attactive TSTO design appears to be a first-stage structure made of only tanks and a second-stage structure made of engines, tanks, electronics, and so on.Improvements in engine specific impulse and spacecraft structural factor are desirable and crucial for SSTO feasibility; indeed, aerodynamic improvements do not yield significant improvements in payload weight.For SSTO configurations, the maximum payload weight behaves almost linearly with respect to the engine specific impulse and the spacecraft structural factor. The same property holds for TSTO configurations as long as the ratio of the structural factors of Stage 2 and Stage 1 is held constant. With reference to the specific impulse/structural factor domain, this property leads to the construction of a zero-payload line separating the feasibility region (positive payload) from the unfeasibility region (negative payload).     

Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient.The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit.In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.     

Optimal sampling and curve interpolation via wavelets

We propose a wavelet-based method for determining optimal sampling positions and inferring underlying functions based on the samples when it is known that the underlying function is Lipschitz. We first propose a Lipschitz regularity-based statistical model for data which are sampled from a Lipschitz curve. And then we propose a wavelet-based interpolation method for generating a Lipschitz curve given a set of points, and derive the optimal sampling positions.     

Optimal Control via Asynchronous Communication Channels

The paper is directed toward the development of a new chapter of control theory that deals with networked systems in which control and communication issues are combined together and in which the delays and limitations of the communication channels between sensors, actuators, and controllers are taken into account. We consider a situation where a single decision maker receives the sensor data and at the same time controls many linear discrete-time partially-observed subsystems perturbed by white noises via randomly delayed communication channels with finite capacities. Neither these delays nor their statistics are known in advance, but each message transmitted is equipped with a time stamp indicating the beginning time of the transfer. Under certain assumptions, a finite-horizon linear-quadratic optimal control problem is solved.     

Optimal control problems via a direct method

Summary In this paper we deal with the minimization problem of a cost functional associated to a nonlinear boundary value control problem of a general form, defined in the fixed time interval [0, 1]. Specifically, we first give conditions which ensure that the nonlinear boundary value control problem is solvable and we study the structure of the relative solution set. Then, based on the properties of this set, we establish conditions ensuring both the existence of quasisolutions and that of solutions of the minimization problem under consideration. Such conditions will depend also on the choice of the control space L^r([0, 1],R ^m) where 1 r + Partially supported by the research project M.P.I. (40%) «Teoria del controllo dei sistemi dinamici».     

Optimal Control Problems via Exact Penalty Functions

The nonsmoothness is viewed by many people as at least an undesirable (if not unavoidable) property. Our aim here is to show that recent developments in Nonsmooth Analysis (especially in Exact Penalization Theory) allow one to treat successfully even some quite smooth problems by tools of Nonsmooth Analysis and Nondifferentiable Optimization. Our approach is illustrated by one Classical Control Problem of finding optimal parameters in a system described by ordinary differential equations.     

Optimal Control of a Chemical Vapor Deposition Reactor

We study a simple model of chemical vapor deposition on a silicon wafer. The control is the flux of chemical species, and the objective is to grow the semiconductor film so that its surface attains a prescribed profile as nearly as possible. The surface is spatially fast oscillating due to the small feature scale, and therefore the problem is formulated in terms of its homogenized approximation. We prove that the optimal control is bang-bang, and we use this information to develop a numerical scheme for computing the optimal control.     

Aggregate Wind Power Production via Coalitional Games and Optimal Control

In this paper, benefits from aggregating independent wind power producers are analyzed in a scenario, in which the producers willingly form coalitions to increase their expected profits. For every deviation from the declared contract, the coalition is penalized and a cost is paid, if the producers want to update their contract. The underlying idea is that coalitions reduce the risk of being penalized. The main contribution of this paper is a new market model and an allocation mechanism based on optimal control and coalitional games with transferable utilities. Optimal control is used to obtain the optimal contract size, while coalitional games provide an insight on stable revenue allocations, namely allocations, that make the grand coalition preferable to all producers.     

Transfers,contracts and strategic games

This paper analyses the role of transfer payments and strategic contracting within two-person strategic form games with monetary payoffs. First, it introduces the notion of transfer equilibrium as a strategy combination for which individual stability can be supported by allowing the possibility of transfers of the induced payoffs. Clearly, Nash equilibria are transfer equilibria, but under common regularity conditions the reverse is also true. This result typically does not hold for finite games without the possibility of randomisation, and transfer equilibria for this particular class are studied in some detail.     

Optimal securitization of credit portfolios via impulse control

We study the optimal loan securitization policy of a commercial bank which is mainly engaged in lending activities. For this we propose a stylized dynamic model which contains the main features affecting the securitization decision. In line with reality we assume that there are non-negligible fixed and variable transaction costs associated with each securitization. The fixed transaction costs lead to a formulation of the optimization problem in an impulse control framework. We prove viscosity solution existence and uniqueness for the quasi-variational inequality associated with this impulse control problem. Iterated optimal stopping is used to find a numerical solution of this PDE, and numerical examples are discussed.     

Optimal domain decomposition via -median methodology using ACO and hybrid ACGA

In this paper an efficient method is developed for decomposing large-scale finite element meshes. A weighted incidence graph is used to transform the connectivity properties of finite element models into those of graphs. A graph G_c of manageable size is obtained from the main graph model by a coarsening algorithm. The p-medians of this graph are selected using two approaches. The first algorithm uses an ant colony optimization and the second algorithm employs a hybrid ant colony together with genetic algorithm. Here, p is the number of subdomains which the finite element meshes is intended to be decomposed. Once the medians are obtained, the nodes in G_c associated with each median are selected. In an expansion process, the nodes of the subdomains in G are obtained. The capabilities of both ant colony optimization, and hybrid ant colony and genetic algorithm are evaluated using many examples of different topology.     

Optimal control of linear time-varying systems via Fourier series

The optimal control of linear time-varying systems with quadratic cost functional is obtained by Fourier series approximation. The properties of Fourier series are first briefly presented and the operational matrix of backward integration together with the product operational matrix are utilized to reduce the optimal control problem to a set of simultaneous linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.     

Optimal Control of Linear Time-Varying Systems via Haar Wavelets

This paper introduces the application of Haar wavelets to the optimal control synthesis for linear time-varying systems. Based upon some useful properties of Haar wavelets, a special product matrix, a related coefficient matrix, and an operational matrix of backward integration are proposed to solve the adjoint equation of optimization. The results obtained by the proposed Haar approach are almost the same as those obtained by the conventional Riccati method.     

Stability Analysis of Optimal Control Systems via Liapunov Method

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Optimal decision trees for categorical data via integer programming

Journal of Global Optimization - Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However,...     

Optimal control of Volterra integral equations via triangular functions

This paper presented an approximate method for solving optimal control problem of Volterra integral equations. The method is based upon orthogonal triangular functions. The error estimates and associated theorems have been proved for optimal control and cost functionals. Some numerical examples illustrate the efficiency of the proposed method.     

Optimal trajectories for quadratic variational processes via invariant imbedding

Consider minimizing the integral $$I = \int_0^T {[\dot w^2 + g(y)w^2 ] dy}$$ where $$w = w(y), \dot w = dw/dy, w(T) = 1, w(0) = free$$ ForT sufficiently small, it is shown that $$w_{opt} = x(t,T), 0 \leqslant t \leqslant T$$ where the functionx, viewed as a function ofT, is a solution of the Cauchy problem $$\begin{gathered} x_T (t,T) = r(T)x(t,T), T \geqslant t \hfill \\ x(t,t) = 1 \hfill \\\end{gathered}$$ and the auxiliary functionr satisfies the Riccati system $$\begin{gathered} r_T = ---g(T) + r^2 , T \geqslant 0 \hfill \\ r(0) = 0 \hfill \\\end{gathered}$$ In the derivation of the Cauchy problem, no use is made of Euler equations, dynamic programming, or Pontryagin's maximum principle. Only ordinary differential equations are employed. The Cauchy problem provides a one-sweep integration procedure; it is intimately connected with the theory of the second variation.     

Optimal control of delay systems via block pulse functions

The concept of coefficient shift matrix is introduced to represent delay variables in block pulse series. The optimal control of a linear delay system with quadratic performance index is then studied via block pulse functions, which convert the problems into the minimization of a quadratic form with linear algebraic equation constraints. The solution of the two-point boundary-value problem with both delay and advanced arguments is circumvented. The control variable obtained is piecewise constant.