Liouville properties and critical value of fully nonlinear elliptic operators

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.Comment: 18 pp, to appear in J. Differential Equation

Pengembangan Multimedia Berbasis Komputer untuk Pembelajaran Matematika bagi Siswa SMA

Tujuan penelitian ini adalah menghasilkan multimedia berbasis komputer yang layak untuk pembelajaran matematika bagi siswa SMA, ditinjau dari aspek pembelajaran, aspek materi, aspek media dan persentase siswa yang mencapai ketuntasan belajar setelah menggunakan produk multimedia pembelajaran hasil pengembangan. Pengembangan multimedia pembelajaran ini dilakukan melalui tahap analisis, desain, produksi, dan uji coba. Tahap analisis meliputi menganalisis kebutuhan siswa, memilih dan menetapkan materi pokok yang akan dikembangkan, dan mengembangkan alat evaluasi. Tahap desain meliputi pembuatan storyboard dan flowchart. Tahap produksi meliputi pengemasan produk awal ke dalam bentuk CD. Tahap uji coba terdiri uji oleh ahli materi dan ahli media, dan uji lapangan meliputi: uji coba satu lawan satu, uji coba kelompok kecil, dan uji coba kelompok besar. Subjek coba terdiri tiga orang siswa untuk uji coba satu lawan satu, 12 orang siswa untuk uji coba kelompok kecil, dan 30 orang siswa untuk uji coba kelompok besar. Data dari hasil uji coba dikumpulkan menggunakan kuesioner dan tes, dan dianalisis dengan statistik deskriptif. Hasil uji coba digunakan untuk memperbaiki multimedia pembelajaran. Hasil penelitian menunjukkan bahwa produk multimedia pembelajaran hasil pengembangan termasuk dalam kategori ”sangat baik”. Hal ini ditinjau dari aspek pembelajaran dengan skor rata-rata 4,45, aspek materi dengan skor rata-rata 4,48, dan aspek media dengan skor rata-rata 4,42. Persentase siswa yang mencapai ketuntasan belajar setelah menggunakan multimedia pembelajaran hasil pengembangan sebanyak 77% dan termasuk dalam kategori ”baik”. Hal tersebut menunjukkan bahwa multimedia pembelajaran matematika yang dikembangkan layak digunakan sebagai media pembelajara

Comparison Principles for subelliptic equations of Monge-Ampere type

We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group

Linear-Quadratic NN-person and Mean-Field Games with Ergodic Cost

We consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $N$ Hamilton-Jacobi-Bellman and $N$ Kolmogorov-Fokker-Planck partial differential equations. We give necessary and sufficient conditions for the existence and uniqueness of quadratic-Gaussian solutions in terms of the solvability of suitable algebraic Riccati and Sylvester equations. Under a symmetry condition on the running costs and for nearly identical players we study the large population limit, $N$ tending to infinity, and find a unique quadratic-Gaussian solution of the pair of Mean Field Game HJB-KFP equations. Examples of explicit solutions are given, in particular for consensus problems.Comment: 31 page

Mean Field Games models of segregation

This paper introduces and analyses some models in the framework of Mean Field Games describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of Mean Field Game theory, in the stationary and in the evolutive case. Numerical methods are proposed, with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.Comment: 35 pages, 10 figure

Modelling and control of a waste to energy plant : waste bed temperature control using a feedback control law

In this dissertation the waste incineration process has been described, an overview of the state of the art control methodologies given and a new approach, based on input/output linearization and extremum seeking has been presented. This approach has been tested on a model appositely designed. The results have shown that it is possible to control the waste bed temperature to certain reference values, with robustness against changes in the waste composition. It is furthermore possible to identify reference values for the waste bed temperature such as the steam ow rate is maximized, while at the same time fulfilling operational constraints