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.     

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.     

Time periodic solutions of nonlinear wave equations

Existence and regularity of solutions of $$(1)u_{tt} - u_{xx} = \varepsilon K(x,t,u,u_t )0< x< \pi ,0 \leqslant t \leqslant 2\pi $$ together with the periodicity and boundary conditions $$(2)u(x,t + 2\pi ) = u(x,t),u(0,t) = 0 = u(\pi ,t)$$ is studied both with an without the dissipation u_t. A solution is a pair (χ, u). A main feature of interest here is an infinite dimensional biofurcation problem. Under appropriate conditions on K, global existence results are obtained by a combination of analytical and topological methods.     

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.     

On the convolution equation with positive kernel expressed via an alternating measure

We consider the integral convolution equation on the half-line or on a finite interval with kernel $$K(x - t) = \int_a^b {e^{ - \left| {x - t} \right|s} d\sigma (s)} $$ with an alternating measure dσ under the conditions $$K(x) > 0, \int_a^b {\frac{1}{s}\left| {d\sigma (s)} \right| < + \infty } , \int_{ - \infty }^\infty {K(x)dx = 2} \int_a^b {\frac{1}{s}d\sigma (s) \leqslant 1} .$$ The solution of the nonlinear Ambartsumyan equation $$\varphi (s) = 1 + \varphi (s) \int_a^b {\frac{{\varphi (p)}}{{s + p}}d\sigma (p)} ,$$ is constructed; it can be effectively used for solving the original convolution equation.     

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.     

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   

On minimum norm solutions

This note investigates the problem $$\min x_p^p /p,s.t.Ax \geqslant b,$$ where 1<p<∞. It is proved that the dual of this problem has the form $$\max b^T y - A^T y_q^q /q,s.t.y \geqslant 0,$$ whereq=p/(p?1). The main contribution is an explicit rule for retrieving a primal solution from a dual one. If an inequality is replaced by an equality, then the corresponding dual variable is not restricted to stay nonnegative. A similar modification exists for interval constraints. Partially regularized problems are also discussed. Finally, we extend an observation of Luenberger, showing that the dual of $$\min x_p ,s.t.Ax \geqslant b,$$ is $$\max b^T y,s.t.y \geqslant 0,A^T y_q \leqslant 1,$$ and sharpening the relation between a primal solution and a dual solution.     

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 $$k(t, y) = min(t, y)$$      

Estimation of the solutions of the Sturm-Liouville equation

Exact estimates are presented for the solutions of the problem $\ddot y + \lambda ^2 p(t)y = 0, y(0) = 0, \dot y(0) = 1$ withp(t) satisfying one of the following conditions: $$(i) |p(t)| \leqslant M< \infty ; (ii) 0< \omega _1 \leqslant p(t) \leqslant \omega _2< \infty ; (iii) \mathop {sup}\limits_x \int_x^{x + T} {p(t)dt = P_T /T.} $$ The extremal solutions are found.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

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 $$\pi (x) = \sum\limits_{p \leqslant x} 1$$ .     

Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.     

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.     

Finite-difference method for solving the first boundary-value problem for a second-order nonlinear ordinary differential equation with a divergent principal part

The approximation is studied of the first boundary-value problem for the equation (1) $$- \frac{d}{{dx}}K(x,\frac{{du}}{{dx}}) + f(x,u) = 0,0< x< 1,$$ with boundary conditions (2) $$u(0) = u(1) = 0$$ by difference boundary-value problems of form (3) $$- \left[ {a(x,w_{\bar x} )} \right]_x + \varphi (x,w) = 0,x \in w_r ,$$ (4) $$w(0) = w(1) = 0.$$ Theorems are established on the solvability of problem (3), (4). Theorems are proved on uniform convergence and on the order of uniform convergence. Here, as usual, boundedness is not assumed, but just the summability of the corresponding derivatives of the solutions of problem (1), (2). Also considered are singular boundary-value problems of form (1), (2), where uniform convergence with order h is proved under assumption of piecewise absolute continuity of the functionf(x,u(x)).     

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].     

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.     

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

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.