A maximum principle for time-lag control problems with bounded state

A maximum principle is obtained for control problems involving a constant time lag τ in both the control and state variables. The problem considered is that of minimizing $$I(x) = \int_{t^0 }^{t^1 } {L (t,x(t), x(t - \tau ), u(t), u(t - \tau )) dt} $$ subject to the constraints 1 $$\begin{gathered} \dot x(t) = f(t,x(t),x(t - \tau ),u(t),u(t - \tau )), \hfill \\ x(t) = \phi (t), u(t) = \eta (t), t^0 - \tau \leqslant t \leqslant t^0 , \hfill \\ \end{gathered} $$ 1 $$\psi _\alpha (t,x(t),x(t - \tau )) \leqslant 0,\alpha = 1, \ldots ,m,$$ 1 $$x^i (t^1 ) = X^i ,i = 1, \ldots ,n$$ . The results are obtained using the method of Hestenes.     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

Conventional penalty optimization methods

In a recent study, the effects of large penalty constants on Ritz penalty methods based on finite-element approximations used in the solution of the control of a system governed by the diffusion equation were established. The problem involves the selection of the inputu(x, t) so as to minimize the cost $$J(u) = \int_0^1 {\int_0^1 {\left\{ {u^2 (x,t) + z^2 (x,t)} \right\}dx dt,} } $$ subject to the constraint $$\partial z/\partial t = \partial ^2 z/\partial x^2 + u(x,t), 0 \leqslant x,t \leqslant 1,$$ with boundary conditions $$z(0,t) = z(1,t) = 0, 0 \leqslant t \leqslant 1,$$ and the initial state $$z(x,0) = z_0 (x), 0 \leqslant x \leqslant 1.$$ Our results verify that the Ritz penalty method exhibits good convergence properties, although the estimates for the convergence rate are cumbersome. In this paper, a conceptually simple procedure based on the conventional penalty method is presented. Some significant advantages of the method is presented. Some significant advantages of the method are the following. It allows easy estimation of its convergence rate. Furthermore, the multiplier method can be used to accelerate the rate of convergence of the method without essentially allowing the penalty constants to tend to infinity; thus, in this way, it is possible to retain the good convergence properties, an important feature which is often glossed over. The paper provides a clear mathematical analysis of how these advantages can be exploited and illustrated with numerical examples.     

On a system of second order differential equations with periodic impulse coefficients

A thorough investigation of the systemd~2y(x):dx~2 p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤xx_0>0) -η, x_0≤x<2π(η>0)p(x)=p(x 2π),-∞相似文献   

A property of an autonomous minimax optimal control problem

A control system \(\dot x = f\left( {x,u} \right)\) ,u) with cost functional $$\mathop {ess \sup }\limits_{T0 \leqslant t \leqslant T1} G\left( {x\left( t \right),u\left( t \right)} \right)$$ is considered. For an optimal pair \(\left( {\bar x\left( \cdot \right),\bar u\left( \cdot \right)} \right)\) ,ū(·)), there is a maximum principle of the form $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right) = \mathop {\max }\limits_{u \in \Omega \left( t \right)} \eta \left( t \right)f\left( {\bar x\left( t \right),u} \right).$$ By means of this fact, it is shown that \(\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right)\) is equal to a constant almost everywhere.     

Well-posedness of a semilinear heat equation with weak initial data

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation $$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$ with initial data in $\dot L_{r,p} $ is studied. We prove the well-posedness when $$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$ and construct non-unique solutions for $$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$ In the second part the well-posedness of the avove IVP for k=2 with μ₀?H ^s (? ⁿ ) is proved if $$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$ and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].     

Optimal trajectories for quadratic variational processes via invariant imbedding

Consider minimizing the integral $$I = \int_0^T {[\dot w^2 + g(y)w^2 ] dy}$$ where $$w = w(y), \dot w = dw/dy, w(T) = 1, w(0) = free$$ ForT sufficiently small, it is shown that $$w_{opt} = x(t,T), 0 \leqslant t \leqslant T$$ where the functionx, viewed as a function ofT, is a solution of the Cauchy problem $$\begin{gathered} x_T (t,T) = r(T)x(t,T), T \geqslant t \hfill \\ x(t,t) = 1 \hfill \\\end{gathered}$$ and the auxiliary functionr satisfies the Riccati system $$\begin{gathered} r_T = ---g(T) + r^2 , T \geqslant 0 \hfill \\ r(0) = 0 \hfill \\\end{gathered}$$ In the derivation of the Cauchy problem, no use is made of Euler equations, dynamic programming, or Pontryagin's maximum principle. Only ordinary differential equations are employed. The Cauchy problem provides a one-sweep integration procedure; it is intimately connected with the theory of the second variation.     

Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

On a multiplicative function on the set of shifted primes

It is proved that if ?(n) is a multiplicative function taking a valueζ on the set of primes such thatζ ³ = 1,ζ ≠ 1 and? ³(p ^r)=1 forr≥2, then there exists aθ ∈ (0, 1), for which $$|\sum\limits_{p \leqslant x} f (p + 1)| \leqslant \theta \pi (x)$$ , where $$\pi (x) = \sum\limits_{p \leqslant x} 1$$ .     

An integral equation and a representation for a Green's function

The final step in the mathematical solution of many problems in mathematical physics and engineering is the solution of a linear, two-point boundary-value problem such as $$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< x \hfill \\ (0) = 0, \dot u(x) = 0 \hfill \\ \end{gathered} $$ Such problems frequently arise in a variational context. In terms of the Green's functionG, the solution is $$u(t) = \int_0^x {G(t, y, x)g(y) dy} $$ It is shown that the Green's function may be represented in the form $$G(t,y,x) = m(t,y) - \int_y^x {q(s)m(t, s) m(y, s)} ds, 0< t< y< x$$ wherem satisfies the Fredholm integral equation $$m(t,x) = k(t,x) - \int_0^x k (t,y) q(y) m(y, x) dy, 0< t< x$$ and the kernelk is $$k(t, y) = min(t, y)$$      

A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$ for some pointz ⁰∈ω, where $$h_E (z) = \sup \{ u(z):u \in PSH (\Omega ),u \leqslant 0 on E,u \leqslant 1 in \Omega \} ,z \in \Omega ,$$ is the extremal function often studied in complex analysis.     

Schrödinger equation and oscillatory Hilbert transforms of second degree

Let $h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $ ( $(i = \sqrt { - 1;} t,x$ -real variables). It is proved that in the rectangle $D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}$ , the function h satisfies the followingfunctional inequality: $\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle. Variation and complexity of solutions of Schrödinger equation are discussed.     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

Removable singular sets for equations of the form\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)}

The following uniformly elliptic equation is considered: $$\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)} , x \in \Omega \subset R^n ,$$ with measurable coefficients. The function f satisfies the condition $$f(x, u, \nabla u) u \geqslant C|u|^{\beta _1 + 1} |\nabla u|^{\beta _1 } , \beta _1 > 0, 0 \leqslant \beta _2 \leqslant 2, \beta _1 + \beta _2 > 1$$ . It is proved that if u(x) is a generalized (in the sense of integral identity) solution in the domain ΩK, where the compactum K has Hausdorff dimension α, and if \(\frac{{2\beta _1 + \beta _2 }}{{\beta _1 + \beta _2 - 1}}< n - \alpha \) , u(x) will be a generalized solution in the domain ω. Moreover, the sufficient removability conditions for the singular set are, in some sense, close to the necessary conditions.     

Approximation of the function sign x in the uniform and integral metrics by means of rational functions

Estimates are obtained for the nonsymmetric deviations R_n [sign x] and R_n [sign x]_L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e^?n, e^?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).     

Optimal rate of integration and ɛ-entropy of a class of analytic functions

The author considers a class F of analytic functions real in the interval [-1, 1] and bounded in the unit circle. As an estimate of the optimal quadrature error R(n) over the class F it is shown that $$_e - \left( {2\sqrt 2 + \frac{1}{{\sqrt 2 }}} \right)\pi \sqrt n \leqslant R(n) \leqslant e^{ - \frac{\pi }{{\sqrt 2 }}n} .$$ With the additional condition that \(\mathop {max}\limits_{x \in [ - 1,1]}\) ¦f(x)¦?B, an estimate is obtained for the ?-entropy H_?(F): $$\frac{8}{{27}}\frac{{(1n2)^2 }}{{\pi ^2 }} \leqslant \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{H_\varepsilon (F)}}{{\left( {\log \frac{1}{\varepsilon }} \right)^3 }} \leqslant \frac{2}{{\pi ^2 }}(1n2)^2 .$$      

Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain

For an equation of mixed type, namely, $$ \left( {1 - \operatorname{sgn} t} \right)u_{tt} + \left( {1 - \operatorname{sgn} t} \right)u_t - 2u_{xx} = 0 $$ in the domain {(x, t) | 0 < x < 1, ?α < t < β}, where α, β are given positive real numbers, we study the problem with boundary conditions $$ u\left( {0,t} \right) = u\left( {1,t} \right) = 0, - \alpha \leqslant t \leqslant \beta , u\left( {x, - \alpha } \right) - u\left( {x,\beta } \right) = \phi \left( x \right), 0 \leqslant x \leqslant 1. $$ . We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition φ(x).