a maximum entropy framework for nonexponential distributions probability distributions having power law tails are observed in a broad range of social economic and biological systems we describe here a potentially useful common framework we derive distribution functions for situations in which a joiner particle k pays some form of price to enter a community of size where costs are subject to economies of scale maximizing the boltzmann gibbs shannon entropy subject to this energy like constraint predicts a distribution having a power law tail it reduces to the boltzmann distribution in the absence of economies of scale we show that the predicted function gives excellent fits to 13 different distribution functions ranging from friendship links in social networks to protein protein interactions to the severity of terrorist attacks this approach may give useful insights into when to expect power law distributions in the natural and social sciences

unknown time varying input delay compensation for uncertain nonlinear systems abstract a tracking controller is developed for a class of uncertain nonlinear systems subject to unknown time varying input delay and additive disturbances a novel filtered error signal is designed using the past states in a finite integral over a constant estimated delay interval the maximum tolerable error between unknown time varying delay and a constant estimate of the delay is determined to establish uniformly ultimately bounded convergence of the tracking error to the origin the controller development is based on an approach which uses lyapunov krasovskii functionals to analyze the effects of unknown sufficiently slowly time varying input delays a stability analysis is provided to prove ultimate boundedness of the tracking error signals numerical simulation results illustrate the performance of the developed robust controller

neural network adaptive control for a class of nonlinear uncertain dynamical systems with asymptotic stability guarantees in this paper a neuroadaptive control framework for continuous and discrete time nonlinear uncertain dynamical systems with input to state stable internal dynamics is developed the proposed framework is lyapunov based and unlike standard neural network nn controllers guaranteeing ultimate boundedness the framework guarantees partial asymptotic stability of the closed loop system that is asymptotic stability with respect to part of the closed loop system states associated with the system plant states the neuroadaptive controllers are constructed without requiring explicit knowledge of the system dynamics other than the assumption that the plant dynamics are continuously differentiable and that the approximation error of uncertain system nonlinearities lie in a small gain type norm bounded conic sector this allows us to merge robust control synthesis tools with nn adaptive control tools to guarantee system stability finally two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach

the maximum entropy inference of solutions to pdes we use the maximum entropy variational technique to infer solutions to second order partial differential equations we restrict ourselves to problems with dirichlet boundary conditions first we construct a basis of moment functions in terms of which we set up the constraints for a maximum entropy inversion for the inference of solutions to elliptic equations then we extend the scheme to the inference of time dependence in evolution equations this extension is done by solving a small system of first order initial value differential equations for the moments of the solution vector

moving horizon state estimation for nonlinear systems using neural networks moving horizon mh state estimation is addressed for nonlinear discrete time systems affected by bounded noises acting on system and measurement equations by minimizing a sliding window least squares cost function such a problem is solved by searching for suboptimal solutions for which a certain error is allowed in the minimization of the cost function nonlinear parameterized approximating functions such as feedforward neural networks are employed for the purpose of design thanks to the offline optimization of the parameters the resulting mh estimation scheme requires a reduced online computational effort simulation results are presented to show the effectiveness of the proposed approach in comparison with other estimation techniques

asymptotic optimal control of uncertain nonlinear euler lagrange systems a sufficient condition to solve an optimal control problem is to solve the hamilton jacobi bellman hjb equation however finding a value function that satisfies the hjb equation for a nonlinear system is challenging for an optimal control problem when a cost function is provided a priori previous efforts have utilized feedback linearization methods which assume exact model knowledge or have developed neural network nn approximations of the hjb value function the result in this paper uses the implicit learning capabilities of the rise control structure to learn the dynamics asymptotically specifically a lyapunov stability analysis is performed to show that the rise feedback term asymptotically identifies the unknown dynamics yielding semi global asymptotic tracking in addition it is shown that the system converges to a state space system that has a quadratic performance index which has been optimized by an additional control element an extension is included to illustrate how a nn can be combined with the previous results experimental results are given to demonstrate the proposed controllers

solving differential equations by a maximum entropy minimum norm method with applications to fokker planck equations the method of maximum entropy minimum norm is utilized to produce a general method of solving differential equations the technique is a generalization and extension of previous work performed by baker jarvis j math phys 30 302 1989 it is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis a number of simplifications occur in this extended method the lagrange multipliers and solution vector can be solved for explicitly thus eliminating the necessity of solving systems of nonlinear equations for the lagrange multipliers as was required in the previous approach it is shown that the solution obtained is equivalent to a minimum norm approximation the maximum entropy solution of differential equations with fourier moments is shown to be identical to a fourier series solution additionally the new method is applied to solving the random walk and fokker planck equations

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

lyapunov based tracking control in the presence of uncertain nonlinear parameterizable friction modeling and compensation for friction effects has been a topic of considerable mainstream interest in motion control research this interest is spawned from the fact that modeling nonlinear friction effects is a theoretically challenging problem and compensating for the effects of friction in a controller has practical ramifications if the friction effects in the system can be accurately modeled there is an improved potential to design controllers that can cancel the effects whereas excessive steady state tracking errors oscillations and limit cycles can result from controllers that do not accurately compensate for friction a tracking controller is developed in this paper for a general euler lagrange system that contains a new continuously differentiable friction model with uncertain nonlinear parameterizable terms to achieve the semi global asymptotic tracking result a recently developed integral feedback compensation strategy is used to identify the friction effects online assuming exact model knowledge of the remaining dynamics a lyapunov based stability analysis is provided to conclude the tracking and friction identification results experimental results illustrate the tracking and friction identification performance of the developed controller

adaptive neural pd control with semiglobal asymptotic stabilization guarantee this paper proves that adaptive neural plus proportional derivative pd control can lead to semiglobal asymptotic stabilization rather than uniform ultimate boundedness for a class of uncertain affine nonlinear systems an integral lyapunov function based ideal control law is introduced to avoid the control singularity problem a variable gain pd control term without the knowledge of plant bounds is presented to semiglobally stabilize the closed loop system based on a linearly parameterized raised cosine radial basis function neural network a key property of optimal approximation is exploited to facilitate stability analysis it is proved that the closed loop system achieves semiglobal asymptotic stability by the appropriate choice of control parameters compared with previous adaptive approximation based semiglobal or asymptotic stabilization approaches our approach not only significantly simplifies control design but also relaxes constraint conditions on the plant two illustrative examples have been provided to verify the theoretical results

robust adaptive asymptotic tracking of nonlinear systems with additive disturbance this note deals with the tracking control of multiple input multiple output mimo nonlinear parametric strict feedback systems in the presence of additive disturbances and parametric uncertainties for such systems c sup 0 robust adaptive controllers usually cannot ensure asymptotic tracking or even regulation in this work under the assumption the disturbances are c sup 2 with bounded time derivatives we present a c sup 0 robust adaptive control construction that guarantees the tracking error is asymptotically driven to zero

a maximum entropy method for solving frobenius perron operator equations let s 0 1 0 1 be a nonsingular transformation that preserves an absolutely continuous invariant measure with density f in this paper we develop a general algorithm for approximating f using the maximum entropy approach

on the convergence of moment problems we study the problem of estimating a nonnegative density given a finite number of moments such problems arise in numerous practical applications as the number of moments increases the estimates will always converge weak as measures but need not converge weakly in l this is related to the existence of functions on a compact metric space which are not essentially riemann integrable in some suitable sense we characterize the type of weak convergence we can expect in terms of riemann integrability and in some cases give error bounds when the estimates are chosen to minimize an objective function with weakly compact level sets such as the bolzmann shannon entropy they will converge weakly in l when an lp norm 1 p is used as the objective the estimates actually converge in norm these results provide theoretical support to the growing popularity of such methods in practice

maximum entropy approach with higher moments for solving fokker planck equation abstract the maximum entropy method with higher number of moments is used to solve the fokker planck equation an adopted newton method is used to iterate the maximum entropy set of equations the method is used to calculate the probability density function of the fokker planck equation the calculations are carried out for three examples 1 the bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems 2 the malthus verhulst model which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light 3 the black scholes equation used in financial market option pricing although the maximum entropy approach has several advantages it is not convergent at large times and so cannot be used to calculate the steady state solution

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

computing reactive front speeds in random flows by variational principle we study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed we develop this representation into a computational tool for computing the front speeds without resorting to closure approximations we demonstrate that the front speeds depend on flow statistics and topologies in a complex and dramatic manner

a continuous asymptotic tracking control strategy for uncertain nonlinear systems in this note we present a new continuous control mechanism that compensates for uncertainty in a class of high order multiple input multiple output nonlinear systems the control strategy is based on limited assumptions on the structure of the system nonlinearities a new lyapunov based stability argument is employed to prove semiglobal asymptotic tracking

inverse optimal design of input to state stabilizing nonlinear controllers we show that input to state stabilizability as defined by sontag 1989 1995 is both necessary and sufficient for the solvability of a hamilton jacobi isaacs equation associated with a meaningful differential game problem similar to but more general than the nonlinear h sub spl infin problem the significance of the result stems from the fact that constructive solutions to the input to state stabilization problem are available presented in the paper and that as shown here inverse optimal controllers possess margins on input to state stability against a certain class of input unmodeled dynamics rather than completion of squares the main tools in our analysis are legendre fenchel transformations and the general form of young s inequality

initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme in this paper we present an approximate method initial value technique for the numerical solution of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations having a boundary layer at one end left or right point it is motivated by the asymptotic behavior of singular perturbation problems the original problem is reduced to an asymptotically equivalent first order initial value problem by approximating the zeroth order term by outer solution obtained by asymptotic expansion and then this initial value problem is solved by an exponentially fitted finite difference scheme some numerical examples are given to illustrate the given method it is observed that the presented method approximates the exact solution very well for crude mesh size h

integrated design of a linear neuro adaptive controller in the presence of norm bounded uncertainties adaptive neural controllers are often criticised for the lack of clear and easy design methodologies that relate adaptive neural network nn design parameters to performance requirements this study proposes a methodology for the design of an integrated linear adaptive model reference controller that guarantees component wise boundedness of the tracking error within an a priori specified compact domain the approach is based on the design of a robust invariant ellipsoidal set where both the nn reconstruction error and the neuro adaptive control are considered as bounded persistent uncertainties we show that all the performance and control requirements for the closed loop system can be expressed as linear matrix inequality constraints this brings the advantage that feasibility and optimal design parameters can be effectively computed while solving a linear optimisation problem an advantage of the method is that it allows a systematic and quantitative evaluation of the interplay between the design param

reducing degeneracy in maximum entropy models of networks based on jaynes s maximum entropy principle exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data observables however their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking where predictions fail here we show that degeneracy appears when the corresponding density of states function is not log concave which is typically the consequence of nonlinear relationships between the constraining observables exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log concave the effectiveness of the method is demonstrated on examples

brief paper a neural network solution for fixed final time optimal control of nonlinear systems in this paper fixed final time optimal control laws using neural networks and hjb equations for general affine in the input nonlinear systems are proposed the method utilizes kronecker matrix methods along with neural network approximation over a compact set to solve a time varying hjb equation the result is a neural network feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated on an example

solution to boundary value problems using the method of maximum entropy the method of maximum entropy is used to solve a class of linear boundary value problems the method is based on using various moments of the differential equation as constraints when maximizing the entropy various examples are presented and compared to exact solutions for varying numbers of moments it is found that the maximum entropy approximation is in many cases better than a fourier series solution for a given number of expansion terms and moments the method is very general and will find applications in many areas of physics a comparison of the amount of work necessary for the maximum entropy solution versus finite difference techniques is presented and it is found that the maximum entropy technique shows promise as an alternative solution technique

boundary layers and kpp fronts in a cellular flow we study an eigenvalue problem associated with a reaction diffusion advection equation of the kpp type in a cellular flow we obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude a 1 it follows that the minimal pulsating traveling front speed c a satisfies the upper and lower bounds c1a1 4 c a c2a1 4 physically the speed enhancement is related to the boundary layer structure of the associated eigenfunction accordingly we establish an averaging along the streamlines principle for the unique positive eigenfunction

a modified piecewise linear markov approximation of markov operators in this paper we propose a modified piecewise linear markov method for approximating markov operators defined on l 1 0 1 we present numerical results with the new method as compared to those using the original piecewise linear markov method

inverse optimal adaptive control for attitude tracking of spacecraft the attitude tracking control problem of a rigid spacecraft with external disturbances and an uncertain inertia matrix is addressed using the adaptive control method the adaptive control laws proposed in this paper are optimal with respect to a family of cost functionals this is achieved by the inverse optimality approach without solving the associated hamilton jacobi isaacs partial differential hjipd equation directly the design of the optimal adaptive controllers is separated into two stages by means of integrator backstepping and a control lyapunov argument is constructed to show that the inverse optimal adaptive controllers achieve h sub spl infin disturbance attenuation with respect to external disturbances and global asymptotic convergence of tracking errors to zero for disturbances with bounded energy the convergence of adaptive parameters is also analyzed in terms of invariant manifold numerical simulations illustrate the performance of the proposed control algorithms

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

range identification for perspective vision systems in this note a new observer is developed to determine range information and hence the three dimensional 3 d coordinates of an object feature moving with affine motion dynamics or the more general ricatti motion dynamics with known motion parameters the unmeasurable range information is determined from a single camera provided an observability condition is satisfied that has physical significance to develop the observer the perspective system is expressed in terms of the nonlinear feature dynamics the structure of the proposed observer is inspired by recent disturbance observer results the proposed technique facilitates a lyapunov based analysis that is less complex than the sliding mode based analysis derived for recent observer designs the analysis demonstrates that the 3 d task space coordinates of the feature point can be asymptotically identified simulation results are provided that illustrate the performance of the observer in the presence of noise

a novel actor critic identifier architecture for approximate optimal control of uncertain nonlinear systems an online adaptive reinforcement learning based solution is developed for the infinite horizon optimal control problem for continuous time uncertain nonlinear systems a novel actor critic identifier aci is proposed to approximate the hamilton jacobi bellman equation using three neural network nn structures actor and critic nns approximate the optimal control and the optimal value function respectively and a robust dynamic neural network identifier asymptotically approximates the uncertain system dynamics an advantage of using the aci architecture is that learning by the actor critic and identifier is continuous and simultaneous without requiring knowledge of system drift dynamics convergence of the algorithm is analyzed using lyapunov based adaptive control methods a persistence of excitation condition is required to guarantee exponential convergence to a bounded region in the neighborhood of the optimal control and uniformly ultimately bounded uub stability of the closed loop system simulation results demonstrate the performance of the actor critic identifier method for approximate optimal control

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

asymptotic initial value methods for two parameter singularly perturbed boundary value problems for second order ordinary differential equations in this paper some numerical methods are presented for singularly perturbed two point boundary value problems for second order ordinary differential equations with two small parameters multiplying the derivatives these methods are distinguished by the fact that initial value problems and or terminal value problems are constructed deduced from the given boundary value problem and then solved by a fitted operator method further error estimates are derived the methods suggested are simple to use and easy to implement the important aspect is that some of the methods suggested are well suited for parallel computing examples are provided to illustrate the methods

asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure the use of a neural network nn as a feedforward control element to compensate for nonlinear system uncertainties has been investigated for over a decade typical nn based controllers yield uniformly ultimately bounded uub stability results due to residual functional reconstruction inaccuracies and an inability to compensate for some system disturbances several researchers have proposed discontinuous feedback controllers e g variable structure or sliding mode controllers to reject the residual errors and yield asymptotic results the research in this paper describes how a recently developed continuous robust integral of the sign of the error rise feedback term can be incorporated with a nn based feedforward term to achieve semi global asymptotic tracking to achieve this result the typical stability analysis for the rise method is modified to enable the incorporation of the nn based feedforward terms and a projection algorithm is developed to guarantee bounded nn weight estimates

approximate time dependent solutions of partial differential equations the maxent minimum norm approach a new approach to inverse problems with incomplete data the maxent minimum norm scheme was recently introduced by baker jarvis bj this method is based on the application of jaynes s maxent principle to a probability distribution defined on the functional space of all the possible solutions of the problem in the present work we consider the application of these ideas to non equilibrium time dependent systems in the spirit of jaynes s information theory approach to statistical mechanics we focus on the behaviour of a small number of physically relevant mean values by recourse to the bj procedure the equations of motion of those mean values are closed and an approximate description of the system s evolution is obtained it is also shown that the bj method is closely related to curado s case i e q 2 and standard mean values of tsallis nonextensive thermostatistics

inverse optimal stabilization of a rigid spacecraft the authors present an approach for constructing optimal feedback control laws for regulation of a rotating rigid spacecraft they employ the inverse optimal control approach which circumvents the task of solving a hamilton jacobi equation and results in a controller optimal with respect to a meaningful cost functional the inverse optimality approach requires the knowledge of a control lyapunov function and a stabilizing control law of a particular form for the spacecraft problem they are both constructed using the method of integrator backstepping the authors give a characterization of nonlinear stability margins achieved with the inverse optimal control law

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

solving fredholm integral equations via a piecewise linear maximum entropy method we propose a piecewise linear approximation method based on the maximum entropy principle to approximate a nonnegative solution of a fredholm integral equation numerically the theoretical analysis and numerical examples show that our method has a convergence rate of order 2 and it can get more accurate approximations with more moments used without ill condition of the classic maximum entropy approach the method can also be applied to solve fredholm integral equations with singular kernels

robust integral of neural network and error sign control of mimo nonlinear systems this paper presents a novel state feedback control scheme for the tracking control of a class of multi input multioutput continuous time nonlinear systems with unknown dynamics and bounded disturbances first the control law consisting of the robust integral of a neural network nn output plus sign of the tracking error feedback multiplied with an adaptive gain is introduced the nn in the control law learns the system dynamics in an online manner while the nn residual reconstruction errors and the bounded disturbances are overcome by the error sign signal since both of the nn output and the error sign signal are included in the integral the continuity of the control input is ensured the controller structure and the nn weight update law are novel in contrast with the previous effort and the semiglobal asymptotic tracking performance is still guaranteed by using the lyapunov analysis in addition the nn weights and all other signals are proved to be bounded simultaneously the proposed approach also relaxes the need for the upper bounds of certain terms which are usually required in the previous designs finally the theoretical results are substantiated with simulations

computational methods for solving two parameter singularly perturbed boundary value problems for second order ordinary differential equations singularly perturbed two point boundary value problems spbvps for second order ordinary differential equations odes with two small parameters multiplying the derivatives are considered computational methods are suggested in this paper to solve this type of problems one of these methods is fitted operator method fom and the other is bondary value technique bvt in fom we suggest an exponentially fitted finite difference effd scheme throughout the domain of definition of the differential equation and the error estimates are derived in bvt the domain of definition of the differential equation is divided into three subintervals called inner and outer regions then we solve the two parameter spbvp tpspbvp over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner regions and a classical finite difference scheme in the outer region the data for boundary conditions at the transition points are obtained using zero order asymptotic expansion approximation of the solution of the problem the bvt is well suited for parallel computing numerical examples are given to illustrate the method

a maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings let s 0 1 0 1 be a nonsingular transformation such that the corresponding frobenius perron operator p s l 1 0 1 l 1 0 1 has a stationary density f we propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f an advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply newton s method the jacobian matrices are positive definite and tri diagonal the numerical experiments show that the new maximum entropy method is more accurate than the markov finite approximation method which also uses piecewise linear functions provided that the involved moments are known this is supported by the convergence rate analysis of the method

brief intelligent optimal control of robotic manipulators using neural networks the paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using neural networks explicit solutions to the hamilton jacobi bellman h j b equation for optimal control of robotic systems are found by solving an algebraic riccati equation it is shown how neural networks can cope with nonlinearities through optimization with no preliminary off line learning phase required the adaptive learning algorithm is derived from lyapunov stability analysis so that both system tracking stability and error convergence can be guaranteed in the closed loop system the filtered tracking error or critic gain and the lyapunov function for the nonlinear analysis are derived from the user input in terms of a specified quadratic performance index simulation results on a two link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances

accurate solution of the thomasfermi equation using the fractional order of rational chebyshev functions in this paper the nonlinear singular thomasfermi differential equation for neutral atoms is solved using the fractional order of rational chebyshev orthogonal functions frcs of the first kind ftn t l on a semi infinite domain where l is an arbitrary numerical parameter first using the quasilinearization method the equation be converted into a sequence of linear ordinary differential equations ldes and then these ldes are solved using the frcs collocation method using 300 collocation points we have obtained a very good approximation solution and the value of the initial slope y 0 1 5880710226113753127186845094239501095 highly accurate to 37 decimal places

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon