In general, the main goal of the optimalcontrol (or optimalimpulsivecontrol) is to choose the control trajectory so that the state and control trajectories maximize/minimize a cost function, called the objective function J . This function must be defined in such a way that its maximum/minimum represents the optimization purpose. For simple systems (i.e. systems where all signals can be expressed by equation that can be easily differentiated or integrated), there are analytical tools that can be used to solve the optimization problem such as the Maximum Principle for Impulse OptimalControl. In  an example (the oil problem) that explains the maximum principle for impulse optimalcontrol is presented and it was possible to reach an optimal solution. However, when the system is more complex, the use of such tool becomes more difficult and, in some cases, can be almost impossible. Here is where numerical methods appear. So, the objective function is computed and then optimization algorithms (such as SQP or IP) are used to minimize the function. In the most general case, the objective function may not be convex.
The impulsiveoptimalcontrol theory has many real-life applications. For example many control problems in queueing theory, population dynamics, mathematical epidemiology, financial math- ematics etc can be formulated as the impulsively controlled systems: see Hordijk and van der Duyn Schouten (1983), Hou and Wong (2011), Korn (1999), Palczewski and Stettner (2007), Pi- unovskiy (2004), Xiao et al. (2006) and the references therein. Roughly speaking, an impulse (or intervention) means the instant change of the state of the system. This results in discontinuous trajectories, leading to technical difficulties when solving optimalcontrol problems. Neverthe- less, it is possible to adjust the dynamic programming method for such models: see Bardi and Capuzzo-Dolcetta (1997), Bensoussan and Lions (1984), Christensen (2014), Davis (1993), Motta and Rampazzo (1996), Yushkevich (1989). The Pontryagin maximum principle (or the closely related Lagrangian approach) can also be used for solving impulsiveoptimalcontrol problems: see Dufour and Miller (2007), Hou and Wong (2011), Miller and Rubinovich (2003), Taringoo and Caines (2013), Xiao et al. (2006).
The form of transaction costs fully justifies the use of impulsive strategies, since we cannot trade continuously in the presence of a constant term in the cost function. One can see that there is a remarkable difference between classical optimalimpulsivecontrol problems (see ) and the one considered here. Namely, costs of the impulses do not appear as a penalizing term in the reward functional; they are encoded in the set of available controls. This type of problems is widely discussed in the literature. If we take F ≡ 0 and G – a utility function, we obtain a problem of optimal portfolio selection with consumption (see , , ). The integral part of the reward functional appears in various banking and cash management applications (see eg. third chapter of  or ). It measures in particular divergence of the portfolio from a selected benchmark (see , ), proper diversification (see ) or variance (see section 6c for more examples). Frequently function F depends on time, which is achieved in our model by extending the set of economic factors with a deterministic variable denoting time.
These equations for phase and frequency error illustrate that when the loop parameters have been fixed, and a specified step in frequency introduced into the input that the decay times of x^(t) and Xg(t) can still be controlled by applying a control Up. Since |up| = * U, it is observed from equations (2-29) that Up can be switched in a manner, so as to have a direct effect on the values of x^ and Xg.
one coded independent variable were used. Accordingly, both assumptions were confirmed, and then, covariance analysis was done to compare the effect of the independent variable (group membership) on the dependent variable (quality of life) (Senn, 1994; Bonate, 2000). Shapiro-Wilk test was used to examine the normality of data (for samples less than 50) and the results revealed the normality of the data. In the results of tests for consistency, Levin variances were used between the components of quality of life and the results showed that the level statics (F) for quality of life is not significant (P> .05), and this indicates that the variance of these variables between subjects (impulse control group and the control group) did not differ and variances are equal. Also to examine the assumption of homogeneity of variance, box test was used and the results showed that the box test was not significant (P=.444 and F=1.01 and BOX=50.91), and therefore, the presuppositions of difference between the covariance are established
The goal of this paper is to study the robust stability, stabilization, and H ∞ -control of uncertain impulsive systems under more general assumption on state jumping. Suﬃcient conditions for the existence of the solutions to the above problems are derived. More- over, these suﬃcient conditions are all in linear matrix inequality (LMI) formalism, which makes their resolution easy.
In recent years, a great research effort has been made to solve optimization problems governed by Partial Differential Equations (PDEs); see, e.g., - and references therein. In many cases, this research has focused on objective functionals with differentiable L 2 terms and non-smoothness resulted from the presence of control and state constraints. However, more recently, the investigation of L 1 cost functionals has become a central topic in PDE-based optimization -, because they give rise to sparse controls that are advantageous in many applications like optimal actuator placement  or impulse control .
Pontryagin Minimum Principle. We also apply the Discrete Pontryagin Minimum Principle to the model T . So we prove that maximal chemo therapy can be optimal and also that it might not depending on the spectral properties of the matrix A , (see below). In section five we determine an optimal strategy for chemo or immune therapy.
dimensional PDE′s involving multiple stable and unstable modes, so synchronization process is more difficult compared to synchronizing using low dimensional ODE′s. Most of coupling schemes for spatiotemporal synchronization are very difficult to implement experimentally because coupling must be applied at all spatial points simultaneously or some variable of driven system must be reset to new values at specific points in space [43, 44, 45, 46], However, these problems can be solved in impulsive synchronization in which much smaller subset of points are driven impulsively. The complex behavior of spatiotemporal synchronization, and long time consumed to solve PDE′s numerically slow down synchronization process generate problems in implementation. This character of PDE′s represent advantages in masking information for secure communication (e.g., many more frequencies are involved in mask on using PDE) and security of information transmission increased [47, 48, 49], and multichannel spread- spectrum communication become efficient since a large number of informative signals can be transmitted
The problem of foot and mouth disease (FMD) is of serious concern to the livestock sector in most nations, especially in developing countries. This paper presents the formulation and analysis of a deterministic model for the transmission dynamics of FMD through a contaminated environment. It is shown that the key parameters that drive the transmission of FMD in a contaminated environment are the shedding, transmission, and decay rates of the virus. Using numerical results, it is depicted that the host-to-host route is more severe than the environmental-to-host route. The model is then transformed into an optimalcontrol problem. Using the Pontryagin’s Maximum Principle, the optimality system is determined. Utilizing a gradient type algorithm with projection, the optimality system is solved for three control strategies: optimal use of vaccination, environmental decontamination, and a combination of vaccination and environmental decontamination. Results show that a combination of vaccination and environmental decontamination is the most optimal strategy. These results indicate that if vaccination and environmental decontamination are used optimally during an outbreak, then FMD transmission can be controlled. Future studies focusing on the control measures for the transmission of FMD in a contaminated environment should aim at reducing the transmission and the shedding rates, while increasing the decay rate.
This paper concentrates on the application of optimalcontrol theory to highlight some aspects of Turkish economy. First the setup is given for Turkey to grow over the balanced path. Then the optimalcontrol problem is identified. The control and state variables are mentioned. The objective is the maximization of life-time discounted utility of the society through optimal choice of consumption which automatically determines investment. We make use of Bellman’s principle to guarantee optimality. We make necessary assumptions (technical assumptions) to make use of calculus techniques for a solution. Some functions to represent utility and production are specified. I used the econometric techniques to estimate some parameters of the functions to decide upon the optimal level of investment for steady-state in Turkey over the period including 2001 crisis. The corresponding differential equations are obtained as a result of the Hamiltonian defined. The phase diagram is prepared to analyse different trajectories.
In this paper, we study the dynamical behaviors of a two-competitive metapopulation system with impulsivecontrol and focus on the stable coexistence of the superior and inferior species. A Poincaré map is introduced to prove the existence of a periodic solution and its stability. It is also shown that a stably positive periodic solution bifurcates from the semi-trivial periodic solution through a transcritical bifurcation.
Until date, many of the host-pathogen interaction mechanisms during HIV infection and progression to AIDS remain unknown. Mathematical modeling of HIV infection is of interest since there are no adequate animal models to test the efficacy of drug regimes. These models provide the essential tool to capture a set of assumptions and provide new insights into questions that are difficult to answer by clinical or experimental studies. To describe various aspects of the interaction between the human immune system and HIV, a number of mathematical models have been formulated. For instance, modeling of the kinetics of HIV RNA under drug therapy has led to substantial insight into the dynamics and pathogenesis of HIV [14, 6, 16] and the existence of multiple reservoirs that have made eradication of the virus difficult. Wodaz and Nowak  presented the basic model of HIV infection, which contains three state variables: healthy CD4+ T-cells, infected CD4+ T-cells, and concentration of free virus. This model which has been modified offers important theoretical insights into the immune control of the virus, based on treatment strategies, while maintaining a simple structure .
we note that when we tacked a new minimization on x ( δ x T ) the transversely condition in (15) become itself transversely condition in (11), and this means that after this minimization the integer order derivative in the dynamical control system dose not effect on the transversely condition.
5. PROPOSED HYBRID FEC-LMS TECHNIQUE According to the literature, the adaptive filtering methods and FEC techniques perform better in noise reduction for PLC and are suitable to deal with impulsive noise. As visualized in Figure-1, impulsive noise has two variants: Periodic and aperiodic where the periodic part produces higher spikes compared to aperiodic part. As such, periodic impulsive noise causes significant performance degradation of the network, which is hard to mitigate . According to , FEC can only mitigate aperiodic impulsive noise and it cannot mitigate periodic noise fully. However, it can be combined with other noise reduction technique to further mitigate periodic impulsive noise to a great extent. In , an adaptive filter was shown to be able to mitigate the periodic impulsive noise. When the periodic impulsive noise is detected, the adaptive filter filters the signal above a particular threshold frequency. Hence, there is a need for a hybrid FEC-LMS noise reduction technique to mitigate the aperiodic and periodic part fully.
One of the important features of the source data for solving the problem of state prediction is their probabilistic nature. Input measurements are specified not just by a value, but also by variances. This provides additional information for training the neural network. Besides, at the output of the neural network we will have not only the values of the parameters of the steady state, but also the probability distributions for these parameters. This information will allow us to make more adequate control decisions based on the results of forecasting. Therefore, to solve the state prediction problem, the Bayesian ANN was applied. The Bayes by Backprop (BBB) method [5-9] provides a posterior distribution of neural network weights. Using it for recurrent ANN is described in .
space consists of infeasible, or very undesirable, solutions. This observation motivated the development of a method for producing reduced search spaces by pruning, thus allowing standard global optimization techniques to be applied more successfully to the reduced box bounds. The GASP method considers the sequential nature of the problem, as it prunes the search space on a phase by phase basis, and results in important computational savings, with search space reduc- tions greater than six orders of magnitude, thus simplifying significantly the subsequent optimization. The method is based on grid sampling in two dimensions for each leg of the mission, with sequential pruning of the search space. The pruning method has been shown to have polynomial time and space complexity, so that it remains tractable as the number of decision variables increases. However, designing multiple gravity assist missions with no deep-space manoeuvres is limited in scope, since many possible trajectories cannot be considered, and, as practice shows, deep space manoeu- vres are used in real missions. If the problem of multiple gravity assist with deep space manoeuvres could be pruned efficiently, then the computational cost of optimizing such trajectories may be significantly reduced. The introduction of deep space manoeuvres offers the further advantage of providing a reasonable approximation of multiple gravity assist trajectories with low-thrust arcs. If a transfer arc is no more simply ballistic but is shaped by one or more propelled manoeuvres (either impulsive or low-thrust) the number of degrees of freedom increases significantly. Hence, an efficient solution process would have to make use of additional information to reduce the number of possible alternatives (pruning the search space) so reducing the computational cost, and increasing the likelihood of finding good solutions. This paper describes a method for pruning of the search space of multiple gravity assist optimization problems with deep space manoeuvres, for the particular case of powered swingbys. The method can be seen as an extension of the GASP method when deep space manoeuvres are considered. Since the pruned problem would still exhibit multiple local minima, the use of a global optimization method to find optimal solutions on the pruned space is proposed.
empirical correlations [6-10]. However, Aprea and Maiorino  found that the mathematic correlations failed to predict the real rejection pressure in their experiment. Meanwhile, Cecchinato et al.  developed sets of mathematic models to simulate the optimal discharge pressure, and stated that results deviated from the one predicted by current correlations of other researchers. Actually, the optimal discharge pressure often varied with the different heat pump designs, different operation condition and even the aging problem. With a real-time controller through the comparison of the transient COP, the heat pump might work in the high system performance operations. Nevertheless, it will consume much time for a dynamic operation heat pump when a traditional real-time controller was used, resulting in a long time that the pump actually not run at the optimal condition.
The control of chaotic systems is a challenging problem that is both of intrinsic interdisci- plinary interest, and also of obvious importance in relation to applications. In general the presence of noise complicates considerably any analysis of the dynamics. But we describe in what follows how a statistical analysis of fluctuations can be used to solve a problem in the nonlinear energy-optimalcontrol of chaos. We consider the energy-optimal entraining of a nonlinear oscillator from a chaotic attractor to another coexisting attractor. Our ap- proach is based on an analogy between the variational formulations of the deterministic control problem and the problem of fluctuations, based on the concept of the optimal fluc- tuational path. One of the key points is the identification of the optimalcontrol function with the optimal fluctuational force. The solution of the energy-optimalcontrol problem can thus be found to an excellent approximation by building the prehistory probability distribution [Dykman et al., 1992] of fluctuational escape trajectories. We compare the performance of the control function found in this way with some earlier adaptive control algorithms.