Optimal control for nonlinear dynamical system of microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1, 3-propanediol (1, 3-PD), the aim of adding glycerol is to obtain as much 1, 3-PD as possible. So a proper feeding rate is required during the process. Taking the concentration of 1, 3-PD at the terminal time as the performance index and the feeding rate of glycerol as the control function, we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non-stationary control. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space; on the other hand, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis of this approximation is also investigated. Numerical results show that, by employing the optimal control policy, the concentration of 1, 3-PD at the terminal time can be increased considerably.     

Optimal switching control for microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Hence, a proper feed strategy is required during the process. In this paper, we present an optimal switching control model based on our proposed controlled switching system. Some properties of the controlled switching system are obtained. Subsequently, we prove the existence of optimal control. In order to deduce the optimality conditions, we transcribe the optimal switching control model into an equivalent one with fixed switching instants and parameters. Finally, the optimality conditions of the equivalent problem are investigated by calculus of variations.     

A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems

This paper considers a free terminal time optimal control problem governed by nonlinear time delayed system, where both the terminal time and the control are required to be determined such that a cost function is minimized subject to continuous inequality state constraints. To solve this free terminal time optimal control problem, the control parameterization technique is applied to approximate the control function as a piecewise constant control function, where both the heights and the switching times are regarded as decision variables. In this way, the free terminal time optimal control problem is approximated as a sequence of optimal parameter selection problems governed by nonlinear time delayed systems, each of which can be viewed as a nonlinear optimization problem. Then, a fully informed particle swarm optimization method is adopted to solve the approximate problem. Finally, two free terminal time optimal control problems, including an optimal fishery control problem, are solved by using the proposed method so as to demonstrate its applicability.     

Problem of time-optimal transition to balanced growth of a bacterial colony

We consider the time optimization problem for a biological model describing the process of the growth of bacterial cells, more precisely, the problem of transition to balanced in minimum time. By a change of variables, a three-dimensional problem is reduced to a two-dimensional one for which we construct an optimal synthesis and present a complete proof of the optimality. In particular, we show that the optimal control has at most one switching point and construct the switching line of the optimal control. We represent a formula for the computation of the optimal time of transition into the terminal state from an arbitrary initial point.     

Optimal control and optimization algorithm of nonlinear impulsive delay system producing 1,3-Propanediol

According to the controllability of pulse times and the amount of jumps in the states at these times in the process of fed-batch culture producing 1,3-propanediol, this paper proposes a terminal optimal control model, whose constraint condition is the nonlinear impulsive delay system. The existence of optimal control is discussed and an optimization algorithm which is applied to each subinternal over one cycle for this optimal control problem is constructed. Finally, the numerical simulations show that the terminal intensity of producing 1,3-propanediol has been increased obviously.     

Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays

In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.     

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative Lévy process with a final payoff at an exponential terminal time and characterize the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.     

Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture

In this paper, we consider an optimal control problem of microbial fermentation process in which glycerol is converted to 1,3-propanediol by Klebsiella pneumoniae in fed-batch culture. During the period of reaction, the variation of pH value is monitored to determine glycerol replenishment quantity, guaranteeing that microorganism can always keep growing fast under enough nutrition. Every time pH value is lower than seven, the quantity of glycerol added is such that pH value returns seven again. Glycerol is poured into reactor at discrete time instant and the quantity is controllable. The problem is to determine for each discrete time instant the glycerol quantity to add and maximize the final concentration of 1,3-propanediol. We present a controlled explicit nonlinear impulsive dynamical system of fed-batch culture with state independent vector measures as controls and study the existence, uniqueness, boundedness, continuous dependence and Gâteaux differentiability of its solution with respect to controls. We then propose a multiple objective programming model and demonstrate the regularity of cost functionals and weak compactness of admissible control set. Finally we discuss the existence of optimal control and implement a hybrid particle swarm optimization algorithm to solve the model optimally. Computational results are presented on a numerical example.     

Parameter Tuning of Multi-Proportional-Integral-Derivative Controllers Based on Optimal Switching Algorithms

Under the framework of switched systems, this paper considers a multi-proportional-integral-derivative controller parameter tuning problem with terminal equality constraints and continuous-time inequality constraints. The switching time and controller parameters are decision variables to be chosen optimally. Firstly, we transform the optimal control problem into an equivalent problem with fixed switching instants by introducing an auxiliary function and a time-scaling transformation. Because of the complexity of constraints, it is difficult to solve the problem by conventional optimization techniques. To overcome this difficulty, a novel exact penalty function is introduced for these constraints. Furthermore, the penalty function is appended to the cost functional to form an augmented cost functional, giving rise to an approximate nonlinear parameter optimization problem that can be solved using any gradient-based method. Convergence results indicate that any local optimal solution of the approximate problem is also a local optimal solution of the original problem as long as the penalty parameter is sufficiently large. Finally, an example is provided to illustrate the effectiveness of the developed algorithm.     

Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture

The bio-dissimilation of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) is a complex bioprocess due to the multiple inhibitions of substrate and products onto the cell growth. In consideration of the fact that both the inhibition mechanisms of 3-hydroxypropionaldehyde (3-HPA) onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane are still unclear, we consider 72 possible metabolic pathways, and establish a novel mathematical model which is represented by an eight-dimensional nonlinear dynamical system. The existence, uniqueness, continuous dependence of solutions to the system and the compactness of the solution set are explored. On the basis of biological robustness, we give a quantitative definition of robustness index of the intracellular substances. Taking the robustness index of the intracellular substances together with the relative error between the experimental data and the computational values of the extracellular substances as a performance index, a parameter identification model is proposed for the nonlinear dynamical system, in which 43848 continuous variables and 1152 discrete variables are involved. A parallel particle swarm optimization — pathways identification algorithm (PPSO-PIA) is constructed to find the optimal pathway and parameters under various experiments conditions. Numerical results show that the optimal pathway and the corresponding dynamical system can describe the continuous fermentation reasonably.     

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.     

Parameter identification and optimization algorithm in microbial continuous culture

In this study, a novelty mathematical model is established to formulate the continuous culture of glycerol to 1,3-Propanediol (1,3-PD) by Klebsiella pneumoniae, in which the inhibition of 3-hydroxypropionaldehyde (3-HPA) to cells growth and activity of some enzymes (such as glycerol dehydratase (GDHt) and 1,3-PD oxidoreductase (PDOR)), and the passive diffusion and active transport of glycerol and 1,3-PD across cell membrane are all taken into consideration. Taking the mean relative error between the experimental data and calculated values as the performance index, a parameter identification model involving multiple nonlinear dynamic systems is presented. The identifiability of the parameter identification model is also proved. Finally, an improved particle swarm optimization (PSO) algorithm is constructed to find the optimal parameters for the systems under substrate limitation and excess conditions, respectively. Numerical results not only show that the established model can be used to describe the continuous fermentation reasonably, but also the improved PSO algorithm is valid.     

Complex metabolic network of glycerol fermentation by Klebsiella pneumoniae and its system identification via biological robustness

The bioconversion of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) can be characterized by an intricate metabolic network of interactions among biochemical fluxes, metabolic compounds, key enzymes and genetic regulation. Since there are some uncertain factors in the fermentation, especially the transport mechanisms of substances across cell membrane, the metabolic network contains multiple possible metabolic systems. In this paper, we establish a complex metabolic network and the corresponding nonlinear hybrid dynamical system aiming to determine the most possible metabolic system. The existence, uniqueness and continuity of solutions are discussed. We quantitatively describe biological robustness and present a system identification model on the basis of robustness performance. The identification problem is decomposed into two subproblems and a procedure is constructed to solve them. Numerical results show that it is most possible that both glycerol and 1,3-PD pass the cell membrane by active transport coupling with passive diffusion under substrate-sufficient conditions, whereas, under substrate-limited conditions, glycerol passes cell membrane by active transport coupling with passive diffusion and 1,3-PD by active transport.