Optimal Control Problems with Switching Points

The main idea of this report is to give an overview of the problems and difficulties that arise in solving optimal control problems with switching points. A brief discussion of existing optimality conditions is given and a numerical approach for solving the multipoint boundary value problems associated with the first-order necessary conditions of optimal control is presented. Two real-life aerospace optimization problems are treated explicitly. These are altitude maximization for a sounding rocket (Goddard Problem) in the presence of a dynamic pressure limit, and range maximization for a supersonic aircraft flying in the vertical, also in the presence of a dynamic pressure limit. In the second problem singular control appears along arcs with active dynamic pressure limit, which in the context of optimal control, represents a first-order state inequality constraint. An extension of the Generalized Legendre-Clebsch Condition to the case of singular control along state/control constrained arcs is presented and is applied to the aircraft range maximization problem stated above. A contribution to the field of Jacobi Necessary Conditions is made by giving a new proof for the non-optimality of conjugate paths in the Accessory Minimum Problem. Because of its simple and explicit character, the new proof may provide the basis for an extension of Jacobi's Necessary Condition to the case of the trajectories with interior point constraints. Finally, the result that touch points cannot occur for first-order state inequality constraints is extended to the case of vector valued control functions

Analytical Modeling of a Novel Microdisk Resonator for Liquid-Phase Sensing: An All-Shear Interaction Device (ASID)

Extensive research on micro/nanomechanical resonators has been performed recently due to their potential to serve as ultra-sensitive devices in chemical/biosensing. These applications often necessitate liquid-phase sensing, introducing significant fluid-induced inertia and energy dissipation that reduces the resonator’s performance. To minimize the detrimental fluid effects on such devices, a novel microdisk resonator supported by two tangentially-oriented, axially-driven “legs” is investigated analytically and effects of the system parameters on the resonator/sensor performance are explored. Since the device surface vibrates primarily parallel to the fluid-structure interface, it is referred to here as an “all-shear interaction device,” or ASID. Analytical modeling of the ASID includes a single-degree-of-freedom model, in which the leg mass and associated fluid resistance are neglected relative to their disk counterparts, and a generalized continuous-system, multi-modal model, in which inertial and fluid effects are included for the entire structure. The resulting analytical formulae along with the parametric studies predict that ASID designs with slender legs yield a global maximum in the quality factor (Qmax) at a “critical” disk radius approximately twice the device thickness, whereas stiffer legs correspond to Qmax occurring for the axial-mode microcantilever (the no-disk limit of the ASID). Additionally, the highest mass and chemical sensitivities (Sm, Sc) and lowest mass limit of detection (LODm) of an ASID-based sensor correspond to the axial-mode microcantilever limit, whereas the chemical LOD (LODc) has a relative minimum at the critical disk size; thus, the “optimal-Q” disk size may be different than the “optimal-sensing” counterpart. The results also show that utilizing stiffer legs will improve Q, Sm, Sc, LODm, and LODc. The theoretical results show both qualitative and quantitative agreement with existing experimental data on liquid-phase quality factor in heptane and in water, while the corresponding theoretical predictions for the fluid-induced resonant frequency shift (typically \u3c 1%) indicate the effectiveness of this novel design. Moreover, the results suggest that appropriately designed ASIDs are capable of achieving unprecedented levels of liquid-phase quality factor in the 300-500 range or even higher. The new theoretical formulae also enable one to easily map experimental data on ASID performance in one liquid to behaviors in other media without performing additional experiments

GPU Accelerated Approach to Numerical Linear Algebra and Matrix Analysis with CFD Applications

A GPU accelerated approach to numerical linear algebra and matrix analysis with CFD applications is presented. The works objectives are to (1) develop stable and efficient algorithms utilizing multiple NVIDIA GPUs with CUDA to accelerate common matrix computations, (2) optimize these algorithms through CPU/GPU memory allocation, GPU kernel development, CPU/GPU communication, data transfer and bandwidth control to (3) develop parallel CFD applications for Navier Stokes and Lattice Boltzmann analysis methods. Special consideration will be given to performing the linear algebra algorithms under certain matrix types (banded, dense, diagonal, sparse, symmetric and triangular). Benchmarks are performed for all analyses with baseline CPU times being determined to find speed-up factors and measure computational capability of the GPU accelerated algorithms. The GPU implemented algorithms used in this work along with the optimization techniques performed are measured against preexisting work and test matrices available in the NIST Matrix Market. CFD analysis looked to strengthen the assessment of this work by providing a direct engineering application to analysis that would benefit from matrix optimization techniques and accelerated algorithms. Overall, this work desired to develop optimization for selected linear algebra and matrix computations performed with modern GPU architectures and CUDA developer which were applied directly to mathematical and engineering applications through CFD analysis

Positive solutions to indefinite problems: a topological approach

The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0, where g : [0,+ 1e[\u2192[0,+ 1e[ is a continuous nonlinearity and a : [0,T]\u2192R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions. In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1). In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. Then, for the Dirichlet problem, we solve a conjecture by G\uf3mez\u2010Re\uf1asco and L\uf3pez\u2010G\uf3mez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976). In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u'' + \u3bb a(t) g(u) = 0, when both \u3bb>0 and the negative part of a(t) are sufficiently large. We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions. As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics. Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains. Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u'' + cu' + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations

Rate-induced transitions for parameter shift systems

Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, em invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system. Considering weak tracking for a nonautonomous Rossler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor

An augmented lagrangian method for optimal control of continuous time dae systems

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2016.Esta dissertação apresenta um algoritmo para resolver problemas de controle ótimo (OCP) de equações algébrico diferenciais (DAE) com base no método de Lagrangiano aumentado. O algoritmo relaxa as equações algébricas e resolve uma sequência de OCPs de equações diferenciais ordinárias (ODE). Os principais benefícios desta abordagem são dois. Em primeiro lugar, as variáveis de estado e as variáveis algébricas podem ter restrições limitantes, mesmo quando os métodos de solução utilizados são indiretos. Em segundo lugar, através da redução do sistema para um ODE, a representação é mais compacta e o OCP pode ser tratado por métodos computacionalmente mais eficientes. Provas matemáticas apresentadas mostram que o algoritmo converge para o valor do objetivo do OCP original e a violação da equação algébrica relaxada vai para zero. Estas propriedades são confirmadas com experimentos numéricos.Abstract: This dissertation presents an algorithm for solving optimal control problems (OCP) of differential algebraic equations (DAE) based on the augmented Lagrangian method.The algorithm relaxes the algebraic equations and solves a sequence of OCPs of ordinary differential equations (ODE). The major benefits of this approach are twofold. First, the state and algebraic variables can be bound constrained, even when the solution methods are indirect. Second, by reducing the system to an ODE, the representation is more compact and can be handled by computationally efficient methods. Mathematical proofs are developed showing that the algorithm converges to the objective value of the original OCP and the violation of the relaxed algebraic equation goes to zero. These properties are confirmed with numerical experiments