Existence Of Saddle Point Research Articles

Numerical impedance matching via extremum seeking control of single-frequency capacitively coupled plasmas

Abstract Impedance matching is a critical component of semiconductor plasma processing for minimizing the reflected power and maximizing the plasma absorption power. In this work, a more realistic plasma model is proposed that couples lumped element circuit, transmission line, and particle-in-cell (PIC) models, along with a modified gradient descent algorithm (GD), to study the impact of presets on the automatic matching process. The effectiveness of the proposed conceptual method is validated by using a single-frequency capacitively coupled plasma as an example. The optimization process with the electrode voltage and the reflection coefficient as the objective function and the optimized state, including plasma parameters, circuit waveforms, and voltage and current on transmission lines, is provided. These results show that the presets, such as initial conditions and objective functions, are closely related to the automatic matching process, resulting in different convergence speeds and optimization results, proving the existence of saddle points in the matching network parameter space. These findings provide valuable information for future experimental and numerical studies in this field.

The nonlinear Benjamin–Feir instability – Hamiltonian dynamics, discrete breathers and steady solutions

We develop a general framework to describe the cubically nonlinear interaction of a degenerate quartet of deep-water gravity waves in one or two spatial dimensions. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the sidebands. For unidirectional waves, the mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin–Feir instability. Moreover, heteroclinic orbits are found to correspond to discrete, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.

GPI-Based design for partially unknown nonlinear two-player zero-sum games

This article focuses on obtaining the solution of two-player zero-sum games (ZSGs) where the saddle point may not exist. From the standpoints of the upper performance index function (PIF) and lower PIF separately, we seek the solution of two-player differential ZSGs without assuming the strict existence of saddle point. To achieve the objective, we apply the generalized policy iteration (GPI) technique to design the controller. With neural network approximators, the system trajectory data are used to update the approximation of the PIFs. It is established that the upper (lower) PIF sequence is convergent along the iteration axis under the GPI scheme. When the two function sequences converge to equal values, the saddle point is attained, and vice versa. Numerical simulation results conformed to the suitability of the designed algorithm.

Stochastic Recursive Zero-sum Differential Games under Model Uncertainty

In this paper, we study stochastic recursive zero-sum differential game problem where the payoff function is described by the solutions of a class of backward stochastic differential equations with uncertainty parameter , which are used to represent different market conditions. In such case, the existence of saddle points for stochastic differential game problems above-mentioned is proved.

The Birth of a Hidden Attractor Through Boundary Crisis

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.

Theoretical understanding of unsteady flow separation for shear flow past three square cylinders in vee shape using structural bifurcation analysis

The unsteady flow separation of two-dimensional (2-D) incompressible shear flow past three identical square cylinders arranged in vee shape is studied in this paper, using theoretical structural bifurcation analysis based on topological equivalence. Through this analysis, the exact location and time of occurrence of bifurcation points (flow separation points) associated with secondary and tertiary vortices on all cylinders are studied. The existence of saddle points is also studied during primary flow separation. Different gap ratios between the downstream cylinders, s/d = 0.6–3.0 (where s is the gap between cylinders, d is the length of cylinder side) with fixed gap 2d between upstream and downstream cylinders for different shear parameter (K) values ranging from \(K=0.0\) to 0.4 are considered at Reynolds number (Re) 100. In this process, the instantaneous vorticity contours and streakline patterns, center-line velocity fluctuation, phase diagram, lift and drag coefficients are studied to confirm the theoretical results. Computations are carried out by using higher order compact finite difference scheme. Present study mainly investigates the effect of K and gap ratio on unsteady flow separation and vortex-shedding phenomenon. All the computed results very efficiently and very accurately reproduce the complex flow phenomenon. Through this study, many noticeable and interesting results are reported for the first time for this problem.

Zero-sum path-dependent stochastic differential games in weak formulation

We consider zero-sum stochastic differential games with possibly path-dependent volatility controls. Unlike the previous literature, we allow for weak solutions of the state equation so that the players’ controls are automatically of feedback type. In particular, we do not require the controls to be “simple,” which has fundamental importance for the possible existence of saddle-points. Under some restrictions, needed for the a priori regularity of the upper and lower value functions of the game, we show that the game value exists when both the appropriate path-dependent Isaacs condition, and the uniqueness of viscosity solutions of the corresponding path-dependent Isaacs-HJB equation hold. We also provide a general verification argument and a characterisation of saddle-points by means of an appropriate notion of second-order backward SDE.

Set optimization problems of generalized semi-continuous set-valued maps with applications

In this paper, some properties of the lower (upper) semi-continuity for set-valued maps taking values in an abstract pre-ordered set are showed, which are then applied to study the existence of solutions for abstract set optimization problems. Moreover, some relationships between minimal solutions for abstract set-valued optimization problems with the vector and set criteria are given under mild conditions. As applications, existence results of solutions for set optimization problems are applied to obtain the existence of saddle points for the set-valued map taking values in the pre-ordered set with the set criterion and of solutions for vector optimization problems whose image space is a real vector space not necessarily endowed with a topology.

Unmanned Flying Vehicle Trajectory Guaranteeing Control at Approach to the Maneuvering Air Target

An algorithm of unmanned flying vehicle-interceptor trajectory guaranteeing control at the stage of maneuvering target approach as a result of game problem solution was developed. Conditions of a saddle point existence for a mode of differential game under consideration have been examined. The results of modeling, confirming the fact that our guaranteeing control algorithm provides the tactical advantage for the unmanned interceptor vehicle, have been presented.

A weak minima approach to the study of the existence of saddle points of integral functionals

We study of the existence of saddle points of the functional J defined in (1.1) both in the regular case, i.e., if E belongs to (LN(Ω))N, and in the singular one, i.e., if E belongs to (L2(Ω))N.

A free boundary approach to the Rosensweig instability of ferrofluids

We establish the existence of saddle points for a free boundary problem describing the two-dimensional free surface of a ferrofluid which undergoes normal field instability (also known as Rosensweig instability). The starting point consists in the ferro-hydrostatic equations for the magnetic potentials in the ferrofluid and air, and the function describing their interface. The former constitute the strong form for the Euler-Lagrange equations of a convex-concave functional. We extend this functional in order to include interfaces that are not necessarily graphs of functions. Saddle points are then found by iterating the direct method of the calculus of variations and by applying classical results of convex analysis. For the existence part we assume a general (arbitrary) non linear magnetization law. We also treat the case of a linear law: we show, via convex duality arguments, that the saddle point is a constrained minimizer of the relevant energy functional of the physical problem.

Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

In the article we present a general theory of augmented Lagrangian functions for cone constrained optimization problems that allows one to study almost all known augmented Lagrangians for cone constrained programs within a unified framework. We develop a new general method for proving the existence of global saddle points of augmented Lagrangian functions, called the localization principle. The localization principle unifies, generalizes and sharpens most of the known results on existence of global saddle points, and, in essence, reduces the problem of the existence of saddle points to a local analysis of optimality conditions. With the use of the localization principle we obtain first necessary and sufficient conditions for the existence of a global saddle point of an augmented Lagrangian for cone constrained minimax problems via both second and first order optimality conditions. In the second part of the paper, we present a general approach to the construction of globally exact augmented Lagrangian functions. The general approach developed in this paper allowed us not only to sharpen most of the existing results on globally exact augmented Lagrangians, but also to construct first globally exact augmented Lagrangian functions for equality constrained optimization problems, for nonlinear second order cone programs and for nonlinear semidefinite programs. These globally exact augmented Lagrangians can be utilized in order to design new superlinearly (or even quadratically) convergent optimization methods for cone constrained optimization problems.

Stability and Unstability of the Standing Wave to Euler Equations

AbstractIn this paper, we first discuss the well-posedness of linearizing equations, and then study the stability and unstability of the 3-D compressible Euler Equation, by analysing the existence of saddle point. In addition, we give the existence of local solutions of the compressible Euler equation.

Determining the Optimal Strategies for Zero-Sum Average Stochastic Positional Games

We consider a class of zero-sum stochastic positional games that generalizes the deterministic antagonistic positional games with average payoffs. We prove the existence of saddle points for the considered class of games and propose an approach for determining the optimal stationary strategies of the players.

Optimal strategies for two-person normalized matrix game with variable payoffs

This paper considers a two-person zero-sum game model in which payoffs are varying in closed intervals. Conditions for the existence of saddle point for this model is studied in this paper. Further, a methodology is developed to obtain the optimal strategy for this game as well as the range of the corresponding optimal values. The theoretical development is verified through numerical example.

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.

Linear quadratic stochastic integral games and related topics

This paper studies linear quadratic games problem for stochastic Volterra integral equations (SVIEs in short) where necessary and sufficient conditions for the existence of saddle points are derived in two different ways. As a consequence, the open problems raised by Chen and Yong (2007) are solved. To characterize the saddle points more clearly, coupled forward-backward stochastic Volterra integral equations and stochastic Fredholm-Volterra integral equations are introduced. Compared with deterministic game problems, some new terms arising from the procedure of deriving the later equations reflect well the essential nature of stochastic systems. Moreover, our representations and arguments are even new in the classical SDEs case.

Education

In this issue, we present two very different papers written in two very different styles. The first is a survey of the multiple timescales method for approximating solutions to differential equations. Multiple timescale methods are common in the literature and an integral part of many graduate programs. However, like riding a bicycle, you need some practice, experience, and insight to use it properly and have meaningful results. The second is an exposition on the Mountain Pass Lemma and related mathematical ideas underlying the existence of saddle points. Despite its name, the second article is no ordinary hike through the hills. In “Profits and Pitfalls of Timescales in Asymptotics,” author Ferdinand Verhulst presents a survey of multiple timescale methods. A colleague of mine once sarcastically pointed out that a tremendous amount of insight can be gleaned from the observation that in almost all problems, parameters are either larger than one or smaller than one, leading to an asymptotic approximation in one form or another. However, one does not have to look far to find problems where it is hard to handle the resulting asymptotic series using a simple Taylor series. Multiple timescales can resolve these problems, but the challenge remains of how to know what the multiple timescales should be without having special knowledge of the problem. Verhulst does an admirable job presenting the basic ideas behind determining timescales a priori using two basic concepts: normal forms and bifurcation theory. In the former case, one transforms the problem into a simpler expression to reveal underlying timescales. In the latter case, understanding the dynamics of a system in terms of bifurcations reveals the qualitative structure of the solution and therefore the timescales. Thus, the author puts order to a body of knowledge that can often appear to students as a disjoint collection of tricks for special problems. In “Mountain Passes and Saddle Points,” author James Bisgard develops the Mountain Pass Lemma of Ambrosetti and Rabinowitz which specifies sufficient conditions for the existence of saddle points. Beginning with accessible examples of smooth functions $F: R^2 \rightarrow R$, we can think of $F$ as the height of the landscape. The central element of this manuscript is a very clear proof of the Mountain Pass Lemma, which essentially states that if there is a local minimum in a valley surrounded by a mountain range and there is a point somewhere beyond the mountain range that is lower than the local minimum, then with an additional special requirement, it can be shown that there must be a mountain pass (saddle point) somewhere. While it may seem that there should always be a mountain pass without any additional requirements, the authors present some counterexamples early in the paper to show that this is not a trivial issue. (I could not resist the urge to fire up my tablet and explore some of the sample surfaces.) The special requirement is the Palais--Smale condition, which is the seemingly peculiar condition that every sequence $x_n$ having two properties, (1) that the height above these points is bounded and (2) that the $\| \nabla F(x_n) \|$ approaches zero, must have a convergent subsequence. The author goes on to extend the Mountain Pass Lemma to domains of any finite dimension and from there to Hilbert spaces. Finally, the author uses the concepts involved in the proof to develop methods for finding saddle points. In summary, the Education section in this issue has something for everyone. The first offering focuses on methods and techniques and would be ideal for a graduate course on perturbation methods or applied mathematics. The second paper is analytic, anchored to theorems and proofs but having ample discussion. It would find a home in an undergraduate and graduate real analysis course. Both take a fresh look at classic subjects in mathematics and could be used to liven up traditional courses in most undergraduate and graduate programs.

On the existence of saddle points for nonlinear second-order cone programming problems

In this paper, we study the existence of local and global saddle points for nonlinear second-order cone programming problems. The existence of local saddle points is developed by using the second-order sufficient conditions, in which a sigma-term is added to reflect the curvature of second-order cone. Furthermore, by dealing with the perturbation of the primal problem, we establish the existence of global saddle points, which can be applicable for the case of multiple optimal solutions. The close relationship between global saddle points and exact penalty representations are discussed as well.

On Saddle Points in Semidefinite Optimization via Separation Scheme

This paper aims at investigating saddle point conditions for augmented Lagrangian functions for semidefinite optimization problems. By means of the image space analysis, the existence of a saddle point is shown to be equivalent to a regular weak nonlinear separation of two suitable subsets in the image space (IS) associated with the given problem. Especially, three classes of augmented Lagrangians based on smooth spectral penalty functions can be derived, as particular cases, from a nonlinear separation scheme in the IS. Without requiring the strict complementarity, it is proved that, under strong second-order sufficiency conditions, all these augmented Lagrangian functions admit a local saddle point, and their Hessians become positive definite in a neighborhood of a local optimal point of the original problem. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set.