Fast adaptive optics and comparatively slower active optics are cornerstones of modern-day astronomy. Such systems are installed on most current large ground-based observatories in the visible or infrared and are included in the design of all future observatories. Their role is twofold; first, to compensate for astronomical seeing, and second, to correct for design and manufacturing errors, as well as thermal and mechanical distortions. What's more, the science goals of future large space observatories in the visible or infrared rely on active and adaptive optics systems for reaching the required wavefront accuracy and stability, with imminent examples the folded segmented primary mirror of the James Webb Space Telescope (JWST) and the deformable mirrors of the Roman Space Telescope, previously called the Wide Field Infrared Survey Telescope (WFIRST). Besides astronomy, adaptive optics find laser applications, for aberration correction and for beam shaping. This thesis was set to explore methods for controlling deformable mirrors with hysteresis, specifically for controlling unimorph deformable mirrors developed and manufactured at the Photonics Laboratory of the FH Münster University of Applied Sciences in Germany. The technology for manufacturing unimorph deformable mirrors has been developed in the past at the Photonics Laboratory and has been expanded in a series of industrial and research projects, both for astronomical and for laser applications. Unimorph deformable mirrors are a promising technology for adaptive and active optics systems, thanks to their paramount mechanical properties and their versatility. However, their piezoelectric actuators exhibit higher hysteresis than most other actuators. The focus of this thesis lies in accurate and precise wavefront control with unimorph deformable mirrors despite their intrinsic hysteresis.   Hysteresis can be compensated with two different approaches. In the feedforward scheme, a mathematical model of the hysteresis is constructed and its inverse model is used in open-loop to drive the deformable mirror. In the feedback scheme, the wavefront deviation — including the hysteresis influence — is measured by a wavefront sensor and the deformable mirror is controlled in closed-loop. These two approaches can be combined for optimal performance. The open-loop compensation using the Prandtl-Ishlinskii formalism has previously been implemented at the Photonics Laboratory and was found to reduce the hysteresis from 15% to about 2%. Nevertheless, the residual uncompensated hysteresis still limits the performance of optical systems that have to be almost diffraction-limited. This thesis consists of two parts that manifest the two activities carried out during this PhD project. The first is image-based aberration correction using extended scenes. This aspires to complement existing technologies for the wavefront control in future space telescopes using active optics. The second is fast defocus sensing for the implementation of a closed-loop focus-shifter, with potential application in laser micromachining. In the first part, the feedback for controlling the unimorph deformable mirror is generated from the imaging detector. This control should correct for constant or slow-changing effects and is classified as "active optics." We designed and built a testbed to evaluate control strategies for compensating for aberrations generated in a conjugate plane. The image-based wavefront correction is designed as a blind optimization with the following configuration parameters: the merit function, the control domain, and the algorithm. We use a common image-sharpness metric as merit function and study it extensively in the domain of the Zernike modes. We show that for a severely aberrated system, the Zernike modes are not orthogonal to each other with respect to the merit function. This effect, that we call "aberration balancing," means that the performance of wavefront-free adaptive and active optics systems can be improved by adding specific low-order aberrations in the case of uncorrectable high-order aberrations, where the amount of the additional aberration depends on the power spectral density of the spatial frequencies of the object. We use this technique in simulation to show how a moon that was hidden in the halo of its planet comes into sight, by balancing secondary astigmatism 0° with astigmatism 0°; and in experiment to increase the limiting resolution of our testbed, by balancing spherical aberration with defocus. With the knowledge of the merit function landscape, we design the control algorithm to account for valleys, plateaus and the aberration balancing. The algorithm is based on the heuristic hill climbing technique, which minimizes the influence of hysteresis. We compare image-based aberration correction in three different control domains, namely the voltage domain, the domain of the Zernike modes, and the domain of the singular modes of the deformable mirror. We demonstrate a combined control scheme that deals with the residual hysteresis left over by the open-loop compensation and with the high dimensionality of the control domains. Moreover, we experimentally show that the control in the domain of the singular modes of the deformable mirror is advantageous for the correction of random aberration in comparison to the domain of the Zernike modes. In the second part of this thesis, the feedback for controlling the unimorph deformable mirror is generated from an additional sensor. Here, the goal is to perform fast focus control, the simplest kind of beam shaping, that falls into the category of "adaptive optics." Recently, the Photonics Laboratory presented a novel unimorph deformable mirror that allows for dynamic focus shift with an actuation rate of 2 kHz. Because of hysteresis and creep, this mirror has to be operated in closed-loop. In the past, a chromatic confocal sensor measured the displacement of the back side of the mirror and the signal was fed back to a PID controller. In the course of this PhD project, a novel defocus sensor based on an astigmatic detection system has been developed. It has a bandwidth higher than 18 kHz and meets the requirements, with high-frequency performance and noise level comparable to those of the commercial chromatic confocal sensor. This sensor can open the way towards a commercial fast focus-shifter based on this mirror, circumventing the limited bandwidth and the complexity of wavefront sensors.@en