SEMIGRUP KUASI KONTINU KUAT DAN SISTEM KENDALI LINEAR BERGANTUNG WAKTU

SUTRIMA, Prof.Dr.Ch. Rini Indrati, M.Si dan Dr.rer.nat. Lina Aryati, M.S

2019 | Disertasi | DOKTOR MATEMATIKA

Banyak permasalahan terkait fenomena reaksi-transportasi dalam sistem fisika dan sistem biologi yang dapat dimodelkan sebagai masalah Cauchy abstrak bergantung waktu. Keluarga operator evolusi sering digunakan untuk menunjukkan kondisi well-posed (eksistensi, ketunggalan, dan kebergantungan kontinu) dari penyelesaian masalah Cauchy bergantung waktu. Dengan pendekatan keluarga operator evolusi, baik masalah Cauchy abstrak bergantung waktu tipe hiperbolik maupun tipe parabolik memerlukan syarat cukup untuk well-posed yang sangat kuat. Keluarga operator evolusi juga dapat digunakan untuk mengidentifikasi keterstabilkan (stabilizability) dan keterdeteksian (detectability) sistem kendali linear bergantung waktu. Khususnya, dalam masalah Cauchy tidak bergantung waktu dan sistem kendali linear tidak bergantung waktu, peran keluarga operator evolusi digantikan oleh semigrup-C0. Dalam sistem kendali linear tidak bergantung waktu juga telah dianalisis keterkendalian (controllability) dan keterobservasian (observability). Sebagai generalisasi dari semigrup-C0, untuk menganalisis masalah Cauchy bergantung waktu dan sistem kendali linear bergantung waktu dikembangkan teori semigrup kuasi-C0 sebagai alternatif dari keluarga operator evolusi. Seperti pada semigrup-C0, pembangkit infinitesimal semigrup kuasi-C0 berperan penting dalam aplikasinya. Peneliti pendahulu telah meneliti beberapa sifat dasar semigrup kuasi-C0, tetapi belum meneliti sifatsifat lanjut seperti yang dimiliki semigrup-C0, termasuk syarat cukup suatu operator menjadi pembangkit infinitesimal semigrup kuasi-C0. Pengembangan semigrup kuasi-C0 dalam masalah Cauchy bergantung waktu, belum diteliti penyelesaian klasik dan penyelesaian lunak, kondisi well-posed, dan pembangkit infinitesimal dari operator pertubasi bergantung waktu. Terkait kestabilan semigrup kuasi-C0, masih terbuka peluang pengembangan konsep kestabilan yang telah ada. Implementasinya dalam sistem kendali linear bergantung waktu, telah dibangun syarat cukup untuk keterkendalian, meskipun kendali dari operator tidak bergantung waktu. Lebih lanjut, dalam sistem kendali linear bergantung waktu belum diteliti tentang keterobservasian. Pada disertasi ini dikembangkan sifat-sifat semigrup kuasi-C0 dan penerapannya pada masalah Cauchy abstrak bergantung waktu dan pada sistem kendali linear bergantung waktu. Sifat-sifat semigrup kuasi-C0 yang dikembangkan meliputi: sifat-sifat dasar, syarat cukup untuk pembangkit infinitesimal, operator Riesz-spektral bergantung waktu, ruang bagian invarian, dan tipe-tipe kestabilan. Penerapan semigrup kuasi-C0 pada masalah Cauchy abstrak bergantung waktu antara lain: penyelesaian klasik dan penyelesaian lunak, kondisi well-posed, dan pembangkit infinitesimal dari operator pertubasi bergantung waktu. Aplikasi semigrup kuasi-C0 pada sistem kendali linear bergantung waktu meliputi karakterisasi keterkendalian, keterobservasian, keterstabilkan, dan keterdeteksian. Secara khusus, juga dikembangkan sistem Riesz-spektral bergantung waktu dan sistem Sturm-Liouville bergantung waktu. Hasil penelitian disertasi disampaikan secara berurutan dalam uraian berikut. Sifat-sifat semigrup kuasi-C0 yang dikembangkan meliputi: konsep batas pertumbuhan, ketunggalan pembangkit infinitesimal, versi Teorema Hille-Yosida dan Teorema Lumer-Phillips, operator Riesz-spektral bergantung waktu, dan ruang bagian invarian. Tipe-tipe kestabilan dari semigrup kuasi-C0 yang berhasil dikembangkan meliputi: kestabilan seragam, stabil eksponensial, stabil lemah, stabil kuat, dan stabil polinomial. Penerapan dalam masalah Cauchy abstrak bergantung waktu, semigrup kuasi-C0 dapat menjustifikasi penyelesaian klasik dan penyelesaian lunak, ekuivalensi pembangkit infinitesimal dengan kondisi wellposed, dan semigrup kuasi-C0 dengan pembangkit infinitesimal dari operator bergantung waktu. Kontribusi pada sistem kendali linier bergantung waktu, semigrup kuasi-C0 dapat menjelaskan konsep keterkendalian dan keterobservasian, yang meliputi: terkendali eksak, terkendali hampiran, terkendali nol eksak, terkendali nol hampiran, terobservasi eksak, dan terobservasi hampiran. Lebih lanjut, dari setiap semigrup kuasi-C0 dapat dibangun semigrup evolusi. Semigrup evolusi digunakan untuk membangun fungsi transfer dan untuk mengidentifikasi stabil eksponensial seragam, terstabilkan, dan terdeteksi. Sifat terstabilkan lengkap juga berhasil dijustifikasi. Beberapa indikator sistem berhasil diaplikasikan dalam sistem Riesz-spektral bergantung waktu dan sistem Sturm-Liouville bergantung waktu. Hasil penelitian disertasi ini diharapkan memberikan sumbangan pengembangan pada teori semigrup dan penelitian terapan, khususnya bidang fisika terapan dan sistem kendali linear terkait hasil penelitian ini. Hasil penelitian juga memberikan peluang penelitian lanjut tentang semigrup kuasi, masalah Cauchy abstrak bergantung waktu, dan sistem kendali linear bergantung waktu.

Many problems of the reaction-transport phenomena in physics and biology systems are described by time-dependent abstract Cauchy problems. The family of evolution operators is frequently used to indicate the well-posedness (existence, uniqueness, and continuous dependence) of solution of the time-dependent Cauchy problems. By the family of evolution operators, the time-dependent abstract Cauchy problems of both hyperbolic and parabolic types require sufficient conditions for well-posedness which are very strong. The family of evolution operators can also be used to identify stabilizability and detectability of a time-dependent linear control system. In particular, for the time-invariant Cauchy problems and linear control systems cases, the role of the family of evolution operators is replaced by C0-semigroup. Furthermore, in time-invariant linear control systems, controllability and observability have been analyzed by the C0-semigroup. As a generalization of C0-semigroups, it was constructed a C0-quasi semigroup which can be used to analyze the time-dependent Cauchy problems and linear control systems. The quasi semigroup is an alternative to the family of evolution operators. As C0-semigroups, the infinitesimal generator of a C0-quasi semigroup plays an important role in the application. The previous researchers have examined several basic properties of C0-quasi semigroup, but they have not examined the advanced properties as in case of C0-semigroup, including the requirements for an operator to be an infinitesimal generator of a C0-quasi semigroup. As development of C0-quasi semigroups in the time-dependent Cauchy problems, it has not been studied the classic and mild solutions, well-posedness, and an infinitesimal generator of time-dependent perturbed operator. Regarding to the stability of C0-quasi semigroups, there are still opportunities to develop the existing theories. As implementations of the C0-quasi semigroups in the time-dependent linear control systems, the sufficient conditions for controllability have been constructed although the controls are the time-invariant operators. Furthermore, in the time-dependent linear control systems, the observability is not identified yet. In this dissertation, we develop the properties of C0-quasi semigroups and apply them in the time-dependent abstract Cauchy problems and linear control systems. The developed properties of C0-quasi semigroups are basic properties, sufficient conditions for infinitesimal generator, time-dependent Riesz-spectral operator, invariant subspace, and concepts of stabilities. The implementations of C0-quasi semigroups in the time-dependent abstract Cauchy problem include the classical and mild solutions, the well-posedness, and the infinitesimal generators of time-dependent perturbed operators. The applications of C0-quasi semigroups in the time-dependent linear control systems include the characterization of controllability, observability, stabilizability, and detectability. Finally, we also develop the time-dependent Riesz-spectral and Sturm-Liouville systems. The results of this dissertation are given in the following sequential descriptions. The developed properties of C0-quasi semigroups include the concept of growth bound, the uniqueness of infinitesimal generator, the version of Hille-Yosida Theorem and Lumer-Phillips Theorem, the time-dependent Riesz-spectral operators, and the invariant subspace. The stabilities of C0-quasi semigroups that are developed are uniformly stable, exponentially stable, weakly stable, strongly stable, and polynomially stable. The applications in the time-dependent abstract Cauchy problems, C0-quasi semigroups can characterize the classical and mild solutions, the equivalence of the infinitesimal generator with the well-posedness, and the C0-quasi semigroup with an infinitesimal generator of time-dependent perturbed operator. The contributions in the time-dependent linear control systems, C0-quasi semigroups can characterize the controllability and observability including exactly controllable, approximately controllable, exactly null controllable, approximately null controllable, exactly observable, and approximately observable. For any C0-quasi semigroup, we can construct an evolution semigroup. This semigroup is used to construct the transfer function and to justify the uniformly exponential stability, stabilizability and detectability. We also identify the complete stabilizability. Some of the developed dynamic properties are applied in the time-dependent Riesz-spectral and Sturm-Liouville systems. The results of this dissertation are expected to contribute the development in the semigroup theory and applied research, especially the applied physics and the linear control systems. The results of the study also provide opportunities of the advanced research about the quasi semigroups, the time-dependent Cauchy problems and linear control systems.

Kata Kunci : semigrup kuasi, kestabilan, masalah Cauchy abstrak bergantung waktu, sistem kendali linear bergantung waktu, terkendali, terobservasi, terstabilkan, terdeteksi.