Many chemical engineering applications, such as safety verification and parameter estimation, require global optimization of dynamic models. Global optimization algorithms typically require obtaining global bounding information of the dynamic system, to aid in locating and verifying the global optimum. The typical approach for providing these bounds is to generate convex relaxations of the dynamic system and minimize them using a local optimization solver. Tighter convex relaxations typically lead to tighter lower bounds, so that the number of iterations in global optimization algorithms can be reduced. To carry out this local optimization efficiently, subgradient-based solvers require gradients or subgradients to be furnished. Smooth convex relaxations would aid local optimization even more. To address these issues and improve the computational performance of global dynamic optimization, this thesis proposes several novel formulations for constructing tight convex relaxations of dynamic systems. In some cases, these relaxations are smooth. Firstly, a new strategy is developed to generate convex relaxations of implicit functions, under minimal assumptions. These convex relaxations are described by parametric programs whose constraints are convex relaxations of the residual function. Compared with established methods for relaxing implicit functions, this new approach does not assume uniqueness of the implicit function and does not require the original residual function to be factorable. This new strategy was demonstrated to construct tighter convex relaxations in multiple numerical examples. Moreover, this new convex relaxation strategy extends to inverse functions, feasible-set mappings in constraint satisfaction problems, as well as parametric ordinary differential equations (ODEs). Using a proof-of-concept implementation in Julia, numerical examples are presented to illustrate the convex relaxations produced for various implicit functions and optimal-value functions. In certain cases, these convex relaxations are tighter than those generated with existing methods. Secondly, a novel optimization-based framework is introduced for computing time-varying interval bounds for ODEs. Such interval bounds are useful for constructing convex relaxations of ODEs, and tighter interval bounds typically translate into tighter convex relaxations. This framework includes several established bounding approaches, but also includes many new approaches. Some of these new methods can generate tighter interval bounds than established methods, which are potentially helpful for constructing tighter convex relaxations of ODEs. Several of these approaches have been implemented in Julia. Thirdly, a new approach is developed to improve a state-of-the-art ODE relaxation method and generate tighter and smooth convex relaxations. Unlike state-of-the-art methods, the auxiliary ODEs used in these new methods for computing convex relaxations have continuous right-hand side functions. Such continuity not only makes the new methods easier to implement, but also permits the evaluation of the subgradients of convex relaxations. Under some additional assumptions, differentiable convex relaxations can be constructed. Besides that, it is demonstrated that the new convex relaxations are at least as tight as state-of-the-art methods, which benefits global dynamic optimization. This approach has been implemented in Julia, and numerical examples are presented. Lastly, a new approach is proposed for generating a guaranteed lower bound for the optimal solution value of a nonconvex optimal control problem (OCP). This lower bound is obtained by constructing a relaxed convex OCP that satisfies the sufficient optimality conditions of Pontryagin's Minimum Principle. Such lower bounding information is useful for optimizing the original nonconvex OCP to a global minimum using deterministic global optimization algorithms. Compared with established methods for underestimating nonconvex OCPs, this new approach constructs tighter lower bounds. Moreover, since it does not involve any numerical approximation of the control and state trajectories, it provides lower bounds that are reliable and consistent. This approach has been implemented for control-affine systems, and numerical examples are presented. / Thesis / Doctor of Philosophy (PhD)

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Computational biology and bioinformatics for apoptosis have been gaining much momentum due to the advances in computational sciences. Both fields use extensive computational techniques and modeling to mimic real world behaviors. One problem of particular interest is on the study of reachability, in which the goal is to determine if a target state or protein concentration in the model is realizable for a signaling pathway. Another interesting problem is to examine faulty pathways and how a fault can make a previously unrealizable state possible, or vice versa. Such analysis can be extremely valuable to the understanding of apoptosis. However, these analyses can be costly or even impractical for some approaches, since they must simulate every aspect of the model. Our approach introduces an abstracted model to represent a portion of the apoptosis signaling pathways as a finite state machine. This abstraction allows us to apply hardware testing and verification techniques and also study the behaviors of the system without full simulation. We proposed a framework that is tailor-built to implement these verification techniques for the discrete model. Through solving Boolean constraint satisfaction problems (SAT-based) and with guided stimulation (Genetic Algorithm), we can further extract the properties and behaviors of the system. Furthermore, our model allows us to conduct cause-effect analysis of the apoptosis signaling pathways. By constructing single- and double-fault models, we are able to study what fault(s) can cause the model to malfunction and the reasons behind it. Unlike simulation, our abstraction approach allows us to study the system properties and system manipulations from a different perspective without fully relying on simulation. Using these observations as hypotheses, we aim to conduct laboratory experiments and further refine our model. / Master of Science

Considerable research has been directed towards efficient test stimuli generation for Register Transfer Level (RTL) circuits. However, stimuli generation frameworks are still not capable of generating effective stimuli for all circuits. Some of the limiting factors are 1) It is hard to ascertain if a branch in the RTL code is reachable, and 2) Some hard-to-reach branches require intelligent algorithms to reach them. Since unreachable branches cannot be reached by any test sequence, we propose a method to deduce unreachability of a branch by looking for the possible values which a signal can take in an RTL code without explicit unrolling of the design. To the best of our knowledge, this method has been able to identify more unreachable branches than any method published in this domain, while being computationally less expensive. Moreover, some branches require very specific values on input signals in specific cycles to reach them. Conventional symbolic execution can generate those values but is computationally expensive. We propose a cycle-by-cycle restrictive symbolic execution that analyzes only a selected subset of program statements to reduce the computational cost. Our proposed method gathers information from an initial execution trace generated by any technique, to intelligently decide specific cycles where the application of this method will be helpful. This method can hybrid with simulation-based test stimuli generation methods to reduce the cost of formal verification. With this method, we were able to reach some previously unreached branches in ITC99 benchmark circuits. / Master of Science

In the first half of this thesis, a novel approach for k-induction bounded model checking using signal domain constraints and property partitioning for proving unreachability of branches in Verilog RTL code is presented. To do this, it approach uses program slicing with respect to the variables of the property under test to generate small-sized SMT formulas that describe the change of variable values between consecutive cycles. Variable substitution is then used on these variables to generate the formula for the subsequent cycles without traversing the abstract syntax tree of the entire design. To reduce the approximation on the induction step, an addition of signal domain constraints is proposed. Moreover, we present the technique for splitting up the property in question to get a better model of the system. The later half of the thesis is concerned with presenting a technique for doing sequential vector compaction on test set generated during simulation based ATPG. Starting with a compaction framework for storing metadata and about the test vectors during generation, this work presented to methods for findind the solution of this compaction problem. The first of these two methods generate the optimum solution by converting the problem appropriate for an optimization solver. The latter method utilizes a heuristics based approach for solving the same problem which generates a comparable but sub-optimal solution while having magnitudes better time and computational efficiency. / Master of Science / Electronic circuits can be described with languages known a hardware description languages like Verilog. The first part of this thesis is concerned about automatically proving if parts of this code is actually useful or reachable when implemented on and actual circuit. The thesis builds up on a method known as bounded model checking which can automatically prove if a property holds or not for a given system. The key insight is obtained from the fact that various memory elements in a circuit is allowed to be only in a certain range of values during the design process. The later half of this thesis is gear towards generating minimum sized inputs values to a circuit required for testing it. This work uses large sized input values to circuits generated by a previously published tool and proposes a way to make them smaller. This can reduce cost immensely for testing circuits in the industry where even the smallest increase in testing time increases cost of development immensely. There are two such approaches presented, one of which gives the optimum result but takes a long time to run for larger circuits, while the other gives comparable but sub-optimal result in a much more time efficient manner.

This thesis examines Turing patterns and how systems control theory can be used to steer them. The mathematical tools used in this thesis consist of linear algebra, partial differential equations (PDE), numerical simulations of PDEs and systems control theory. These topics will all be briefly introduced. Following that will be a section on the numerical study of Turing patters. The bulk of the thesis will focus on network controllability. The subjects explored are an analytical study of controllability, the simulations of specific patterns using controllability and a numerical analysis of the minimal energy control for a Turing system

La conception d'architectures opérationnelles d'un système de contrôle-commande est une phase très importante lors de la conception de systèmes de production d'énergie. Cette phase consiste à projeter l'architecture fonctionnelle sur l'architecture organique tout en respectant des contraintes de capacité et de sûreté, c'est-à-dire à allouer les fonctions de commande à un ensemble de contrôleurs tout en respectant ces contraintes. Les travaux présentés dans cette thèse proposent : i)une formalisation des données et contraintes du problème d'allocation de fonctions - ii)une méthode d'allocation, par recherche d'atteignabilité, basée sur un mécanisme d'appel/réponse dans un réseau d'automates communicants à variables entières - iii)la comparaison de cette méthode à une méthode de résolution par programmation linéaire en nombres entiers. Les résultats de ces travaux ont été validés sur des exemples de taille réelle et ouvrent la voie à des couplages entre recherche d'atteignabilité et programmation linéaire en nombres entiers pour la résolution de problèmes de satisfaction de systèmes de contraintes non linéaires. / The design of operational control architectures is a very important step of the design of energy production systems. This step consists in mapping the functional architecture of the system onto its hardware architecture while respecting capacity and safety constraints, i.e. in allocating control functions to a set of controllers while respecting these constraints. The work presented in this thesis presents: i) a formalization of the data and constraints of the function allocation problem- ii) a mapping method, by reachability analysis, based on a request/response mechanism in a network of communicating automata with integer variables- iii) a comparison between this method and a resolution method by integer linear programming. The results of this work have been validated on examples of actual size and open the way to the coupling between reachability analysis and integer linear programming for the resolution of satisfaction problems for non-linear constraint systems.

Planejamento em inteligência artificial é a tarefa de determinar ações que satisfaçam um dado objetivo. Nos problemas de planejamento sob incerteza, as ações podem ter efeitos probabilísticos. Esses problemas são modelados como Processos de Decisão Markovianos (Markov Decision Processes - MDPs), modelos que permitem o cálculo de soluções ótimas considerando o valor esperado de cada ação em cada estado. Contudo, resolver problemas grandes de planejamento probabilístico, i.e., com um grande número de estados e ações, é um enorme desafio. MDPs grandes podem ser reduzidos através da computação de bissimulações estocásticas, i.e., relações de equivalência sobre o conjunto de estados do MDP original. A partir das bissimulações estocásticas, que podem ser exatas ou aproximadas, é possível obter um modelo abstrato reduzido que pode ser mais fácil de resolver do que o MDP original. No entanto, para problemas de alguns domínios, a computação da bissimulação estocástica sobre todo o espaço de estados é inviável. Os algoritmos propostos neste trabalho estendem os algoritmos usados para a computação de bissimulações estocásticas para MDPs de forma que elas sejam computadas sobre o conjunto de estados alcançáveis a partir de um dado estado inicial, que pode ser muito menor do que o conjunto de estados completo. Os resultados experimentais mostram que é possível resolver problemas grandes de planejamento probabilístico com desempenho superior às técnicas conhecidas de bissimulação estocástica. / Planning in artificial intelligence is the task of finding actions to reach a given goal. In planning under uncertainty, the actions can have probabilistic effects. This problems are modeled using Markov Decision Processes (MDPs), models that enable the computation of optimal solutions considering the expected value of each action when applied in each state. However, to solve big probabilistic planning problems, i.e., those with a large number of states and actions, is still a challenge. Large MDPs can be reduced by computing stochastic bisimulations, i.e., equivalence relations over the original MDP states. From the stochastic bisimulations, that can be exact or approximated, it is possible to get an abstract reduced model that can be easier to solve than the original MDP. But, for some problems, the stochastic bisimulation computation over the whole state space is unfeasible. The algorithms proposed in this work extend the algorithms that are used to compute stochastic bisimulations for MDPs in a way that they can be computed over the reachable set of states with a given initial state, which can be much smaller than the complete set of states. The empirical results show that it is possible to solve large probabilistic planning problems with better performance than the known techniques of stochastic bisimulation.

<p>Software is finding its way into an increasing range of devices (phones, medical equipment, cars...). A challenge is to design <i>verification</i> methods to ensure correctness of software. </p><p>We focus on <i>model checking</i>, an approach in which an abstract model of the implementation and a specification of requirements are provided. The task is to answer automatically whether the system conforms with its specification.We concentrate on (i) timed systems, and (ii) parameterized systems.</p><p><i>Timed systems </i>can be modeled and verified using the classical model of timed automata. Correctness is translated to language inclusion between two timed automata representing the implementation and the specification. We consider variants of timed automata, and show that the problem is at best highly complex, at worst undecidable.</p><p>A <i>parameterized system</i> contains a variable number of components. The problem is to verify correctness regardless of the number of components. <i>Regular model checking</i> is a prominent method which uses finite-state automata. We present a semi-symbolic minimization algorithm combining the partition refinement algorithm by Paige and Tarjan with decision diagrams.</p><p>Finally, we consider systems which are both timed and parameterized: <i>Timed Petri Nets</i> (<i>TPNs</i>), and <i>Timed Networks</i> (<i>TNs</i>). We present a method for checking safety properties of TPNs based on forward reachability analysis with acceleration. We show that verifying safety properties of TNs is undecidable when each process has at least two clocks, and explore decidable variations of this problem.</p>