PA-28 pilot reported during climb the aircraft elevator controls were resisting inputs and the airspeed indicated incorrectly. Pilot continued to planned destination where maintenance inspection found a defective component of the pitot static system.

Narrative

On Day 0; I was preparing to fly my PA-28 Cherokee to ZZZ with intentions of bringing the plane in for annual service to a shop on the field. I performed a preflight briefing using Foreflight and then a physical checkout of the plane. I reviewed the preflight checklist for completion which includes 'free and clear' of the controls. At just after XA00 I took off on a VFR flight from ZZZ1 on Runway XX. ZZZ1 is an airport which lies under the outer shelf of ZZZ2 class C airspace. After a normal takeoff; I turned to the southwest and continued to climb. At about 1200-1300 ft.; I noticed the elevator controls were resisting my inputs to climb and saw the airspeed reading 23 kts. I knew this was wrong because the plane was flying normally. I considered returning to ZZZ1; but decided against it since there are homes close to the end of Runway XX. Since I was able to continue a climb; I thought it prudent to attempt to resolve the problem or get the plane to an airport with repair or emergency services if needed. I looked at the Garmin GFC 500 auto pilot to see if it was 'on'; but it was indicating 'off.' I turned the flight director and autopilot on and then off but there was no change in the flight controls. I was suspicious ESP (Electronic Stability Protection) was engaged but did not hear any aural alerts or see any error 'X' out messages on the G3X Touch. I considered pulling the Autopilot fuses but decided to 'do no harm.' I was able to maintain altitude by keeping pressure on the elevator controls so I continued the flight.I listened to ATIS at ZZZ; received information; and then called the Tower. I reported the airspeed reading of 23 knots and that I had no elevator control which was not entirely correct. I wish I had taken time to give a better explanation of the elevator resistance to climb. Tower asked if I wanted to [request priority handling] and told Tower I did. I recall Tower asked where I was and requested ident. I performed the ident but I was somewhat flustered at the moment and told Tower I was north. Tower told me I was south (?) of the field and I recall making a comment to Tower their stated location was correct. I could see I was nearing ZZZ3 airspace and flew further to the southwest to avoid conflict. Tower asked if I wanted to land at ZZZ3 and I recall telling Tower about preferring to come to ZZZ because my shop was there and the controls continued to operate in a predictable manner. Tower asked if I could see the airport and I stated I could.I heard ZZZ coordinate traffic for my arrival. In approaching ZZZ; I was able to descend on a normal VFR approach using the PAPI. I was confident in my estimates of airspeed based on my experience in this plane and using visual clues. As I descended; I heard the aural alert indicating 500 AGL. Shortly thereafter I heard a clunk and I could feel that the flight controls were back to normal. I proceeded to land normally. Tower asked where I wanted to go; and I replied I wanted to go to the south ramp which is where my maintenance shop is located. Since this was my first emergency I was listening to hear instructions to call a number but none were given.As I taxied to the ramp I gave the firetrucks a thumbs up and they moved aside and allowed me to proceed to the aircraft maintenance shop. I was met at the maintenance shop by owner/IA (Inspection Authority) who was aware of the [requested priority] and he spent the remainder of the business day verifying there were no flight control issues. At this point everyone involved was convinced this incident was caused by automatic activation of ESP. On Day 3; I went to the avionics shop and spoke to the owner/IA and his technical staff about the [incident]. The avionics shop wanted the plane immediately after the annual inspection which was complete on Day 7. That same day the avionics shop called me to report they had discovered a defective plastic fitting in the pitot static system. The barbed fitting was out of round and was of an uneven thickness that led to the fracture. I also discussed the [communication of the incident] with my CFII who presented a couple possible alternate scenarios. He agreed with my decision to make a report. This report is my recollection of the events with the intent of being a cooperative and safe pilot. Thank you.